Characterizations of Gelfand Rings Specially Clean Rings and Their Dual Rings

Home , Reduced ring

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Introduction 1 A otisauiu minimal unique a contains A uly ring a Dually, . R0 13A99, 4R10, ;purified g; 1,[4], [1], ings. A lly lts is t 2 M.AGHAJANIANDA.TARIZADEH only if its maximal spectrum is the Zariski retraction of its prime spectrum”, see [15, Theorem 1.2]. This result plays a major role in proving some results of our paper. In Theorem 6.2, we prove the dual of this result which states that a ring is a mp-ring if and only if its minimal spectrum is the flat retraction of its prime spectrum. Theorem 6.3 is a further important result which is obtained in this direction. Although mp-rings have been around for some time, there is no substantial account of their characterizations and basic properties in the literature. This may be because Gelfand rings are tied up with the Zariski topology, see Theorem 4.3. By contrast, we show that mp-rings are tied up with the flat topology, see Theorem 6.2. The flat topology is less known than the Zariski topology in the literature. For the flat topology please consider §2 and for more details see [26]. In this paper, we give a coherent account of mp-rings and their basic and sophisticated properties, see Theorems 6.2, 6.3, 6.4 and 7.3. Clean rings, as a subclass of Gelfand rings, are also investigated in this paper. Recall that a ring A is called a clean ring if each element of it can be written as a sum of an idempotent and an invertible element of that ring. This simple definition has some spectacular equivalents, see Theorem 5.1. This result, in particular, tells us that if A is a clean ring then a system of equations fi(x1, ..., xn) = 0 with i =1, ..., d over A has a solution in A provided that this system has a solution in each local ring Am with m a maximal ideal of A. Clean rings have been extensively studied in the literature over the past and recent years, see e.g. [4], [5], [13], [18], [22] and [23]. But according to [5], although ex- amples and constructions of exchange rings abound, there is a pressing need for new constructions to aid the development of the theory (note that in the commutative case, exchange rings and clean rings are the same, see Theorem 5.1). Toward realizing this purpose, then Theorem 5.1 can be considered as the culmination and strengthening of all of the results in the literature which are related to the characterization of commutative clean rings. Then we introduce and study a new class of rings, we call them purified rings. Specifically, in Theorem 8.5 we characterize reduced purified rings. This result, in particular, tells us that if A is a reduced purified ring then a system of equations fi(x1, ..., xn) = 0 over A has a solution in A provided that this system has a solution in each domain A/p with p a minimal prime ideal of A. In [1, Theorems 1.8 and 1.9], the pure ideals of a reduced Gelfand ring are characterized. In Theorem 7.2, we have improved these results, especially a very simple proof is given for [1, Theorem 1.8]. Then in Theorem 7.3 and Corollary 7.4, the pure ideals of a reduced mp-ring GELFAND RINGS, CLEAN RINGS AND THEIR DUAL RINGS 3 are characterized. Corollaries 7.5 and 7.6 are further results which are obtained in this direction. Our paper also contains some geometric results. In particular, we show that every compact scheme is an affine scheme, see Theorem 3.4. It is also shown that the underlying space of a separated scheme or more generally a quasi-separated scheme is Hausdorff if and only if every point of it is a closed point, see Corollary 3.6.

2. Preliminaries Here we recall some material which is needed in the sequel. In this paper all rings are commutative. An ideal I of a ring A is called a pure ideal if the canonical ring map A A/I is a ﬂat ring map. →

Theorem 2.1. Let I be an ideal of a ring A. Then the following statements are equivalent. (i) I is a pure ideal. (ii) Ann(f)+ I = A for all f I. (iii) If f I then there exists∈ some g I such that f(1 g)=0. (iv) I = ∈f A : Ann(f)+ I = A . ∈ − { ∈ }

Proof. See [8, Chap 7, Proposition 2] or [14, Tag 04PS].

If a prime ideal of a ring A is a pure ideal, then it is a minimal prime ideal of A. Pure ideals are quite interesting and have important applications even in non-commutative situations, see [8, Chaps. 7, 8]. If X is a topological space, then by π0(X) we mean the set of connected components of X equipped with the quotient topology. A subspace Y of a topological space X is called a retraction of X if there exists a continuous map γ : X Y such that γ(y) = y for all y Y . Such a map γ is called a retraction→ map. A quasi-compact and∈ Hausdorff topological space is called a compact space. A subset E Spec(A) is said to be stable under the generalization if whenever p ⊆ E and q is a prime ideal of A such that q p, then q E. The∈ dual notion is called stable under the specialization.⊆ If I is∈ a pure ideal of a ring A then by Theorem 2.1, V (I) is stable under the generalization. If an ideal of a ring A is generated by a set of idempotents of A, then it is called a regular ideal of A. Every regular ideal is a pure ideal, but the converse does not necessarily hold. Every maximal element of the set of proper and regular ideals of A is called a max-regular ideal of A. The 4 M.AGHAJANIANDA.TARIZADEH set of max-regular ideals of A is called the Pierce spectrum of A and denoted by Sp(A). It is a compact and totally disconnected space whose basis opens are precisely of the form Uf = M Sp(A): f / M where f A is an idempotent. We also have{ the∈ homeomorphism∈ } π Spec(A∈) Sp(A), for more details see [26]. 0 ≃ If A is a ring, then there exists a (unique) topology over Spec(A) such that the collection of subsets V (f) = p Spec(A): f p with f A forms a sub-basis for the opens{ of∈ this topology. It∈ is} called the∈ flat (or, inverse) topology. Moreover there is a (unique) topology over Spec(A) such that the collection of subsets D(f) V (g) with f,g A forms a sub-basis for the opens of this topology.∩ It is called the patch∈ (or, constructible) topology. If p is a prime ideal of A, then Λ(p) = q Spec(A): q p is the closure of p in Spec(A) with respect to{ the∈ flat topology.⊆ It} can be shown that{ the} flat closed subsets of Spec(A) are precisely of the form Im ϕ∗ where ϕ : A B is a flat ring map. For more details on the flat and patch topologies→ please consider [26]. The following result is due to Grothendieck and has found interesting applications in this paper.

Theorem 2.2. The map f D(f) is a bijective function from the set of idempotents of a ring A onto the set of clopen (both open and closed) subsets of Spec(A).

Proof. See [14, 00EE].

Let ϕ : A B be a morphism of rings. We say that the idempotents of A can be→ lifted along ϕ if e B is an idempotent, then there exists an idempotent e′ A such that∈ϕ(e′)= e. ∈

Theorem 2.3. Let X be a compact and totally disconnected topological space. Then the set of clopens of X forms a basis for the topology of X. If moreover, X has an open covering C with the property that every open subset of each member of C is again a member of C , then there exist a finite number W1, ..., Wq C of pairwise disjoint clopens of X q ∈ such that X = Wk. kS=1 Proof. The first part of the assertion is deduced from [16, Theorem 6.1.23], and the second part follows from the first part. GELFAND RINGS, CLEAN RINGS AND THEIR DUAL RINGS 5

3. Maximality of primes Lemma 3.1. Let S and T be multiplicative subsets of a ring A. Then S−1A T −1A =0 iﬀ there exist f S and g T such that fg =0. ⊗A ∈ ∈

−1 −1 −1 Proof. Suppose S A A T A = 0. Setting M := S A. Then −1 −1 ⊗ T M M A T A = 0. Thus the image of the unit of M under the canonical≃ map⊗ M T −1M is zero. Hence, there exists some g T such that g/1 = 0→ in M. Thus there is some f S such that fg =∈ 0. ′ ′ −∈1 −1 Conversely, if r/s r /s is a pure tensor of S A A T A then we may write r/s r′⊗/s′ = rf/sf r′g/s′g = r/sf r⊗′fg/s′g = 0. ⊗ ⊗ ⊗

Lemma 3.2. If p and q are distinct minimal prime ideals of a ring A, then there exist f A p and g A q such that fg =0. ∈ \ ∈ \

Proof. It suﬃces to show that 0 S := (A p)(A q). If 0 / S then there exists a prime ideal p′ of A∈such that p\′ S =\ . This yields∈ that p = p′ = q. But this is a contradiction. ∩ ∅

2 Let A be a ring. Consider the relation S = (p, q) X : Ap A Aq = 0 on X = Spec(A). By Lemma 3.1, (p, q) {S iff 0∈/ (A p)(⊗A q6). This} relation is reflexive and symmetric. Let∈ be∈ the\ equivalence\ ∼S relation generated by S. Thus p S q if and only if there exists a finite set p , ..., p of prime ideals of A∼ with n > 2 such that p = p, p = q { 1 n} 1 n and Api A Api+1 = 0 for all 1 6 i 6 n 1. Note that it may happen that p ⊗ q but A6 A = 0. Also note− that in Theorem 3.3 (x), by ∼S p ⊗A q [p] we mean the equivalence class of S containing p. In the following result, the criteria∼ (i)-(iv) are well known, and the remaining are new. The Zariski, flat and patch topologies on Spec(A) are denoted by , and , respectively. Z F P

Theorem 3.3. For a ring A the following statements are equivalent. (i) Every prime ideal of A is a maximal ideal. (ii) is Hausdorff. (iii)Z = . (iv) A/Z NPis absolutely flat where N is the nil-radical of A. (v) If p and q are distinct prime ideals of A, then there exist f A p and g A q such that fg =0. ∈ \ (vi) ∈is Hausdorff.\ (vii)FEvery flat epimorphism of rings with source A is surjective. (viii) If p is a prime ideal of A, then the canonical map π : A A p → p 6 M.AGHAJANIANDA.TARIZADEH is surjective. (ix) = . (x) IfZp isF a prime ideal of A, then [p]= p . { }

Proof. For (i) (ii) see [17, Theorem 3.6], for (i) (iii) see [9, Theorem 2.1], and⇔ for (i) (iv) see [20, Theorem 3.1]. ⇔ (i) (v) : It follows from⇔ Lemma 3.2. (v)⇒ (i) : It is straightforward. (i) ⇒ (vi) : If p and q are distinct prime ideals of A then by the hypothesis,⇒ p + q = A. Thus there are f p and g q such that f + g = 1. Therefore V (f) V (g)= . ∈ ∈ (vi) (i): If m is a maximal∩ ideal of∅A, then Λ(m)= m , because in a Hausdorff⇒ space each point is a closed point. This shows{ } that every prime ideal of A is a maximal ideal. (i) (vii) : Let ϕ : A B be a flat epimorphism of rings. It suffices ⇒ → to show that the induced morphism ϕp : Ap Bp is surjective for all p Spec(A). If pB = B, then we show that→ϕ is an isomorphism of ∈ 6 p rings. Clearly ϕp is a flat epimorphism of rings, because both flat ring maps and epimorphisms of rings are stable under base change. Also, Bp as Ap module is faithfully flat, since pB = B. Hence, ϕp is an isomorphism− of rings, since it is well known that6 every faithfully flat epimorphism of rings is an isomorphism. But if pB = B, then we show that Bp Ap A B =0. If Ap A B = 0 then it has a prime ideal P , and so in≃ the following⊗ pushout⊗ diagram:6 ϕ A / B

π µ A λ / A B p p ⊗A −1 we have λ (P )= pAp, since by the hypothesis every prime ideal of A is a maximal ideal. Thus p = ϕ−1(q) where q := µ−1(P ). It follows that pB q = B which is a contradiction. (vii) (⊆viii)6 : There is nothing to prove. (viii)⇒ (i) : For each f A p there exists some g A such that ⇒ ∈ \ ∈ g/1=1/f in Ap. Thus there exists an element h A p such that h(1 fg) = 0. It follows that 1 fg p and so A/∈p is a\ field. (vi)− (ix) : If X = Spec(A) then− the∈ map (X, ) (X, ) given by x ⇒ x is continuous, since the patch topology isP finer→ thanF the flat topology. It is also a closed map, because the space (X, ) is quasi- compact and (X, ) is Hausdorff. Hence, it is a homeomorphism.P Thus = . Using (iii),F then we get that = . F P Z F GELFAND RINGS, CLEAN RINGS AND THEIR DUAL RINGS 7

(ix) (i) : If p is a prime of A, then V (p) = Λ(p) and so p is a maximal⇒ ideal. (i) (x) : Let m be a maximal ideal of A and m′ [m]. Thus there ⇒ ∈ exists a ﬁnite set m1, ..., mn of maximal ideals of A with n > 2 such { ′ } that m1 = m, mn = m and Ami A Ami+1 = 0 for all 1 6 i 6 n 1. ⊗ 6 ′ − Thus by Lemma 3.2, m = m1 = ... = mn = m . (x) (i) : Let p be a prime of A. There is a maximal ideal m of A ⇒ such that p m. By Lemma 3.1, Ap A Am = 0. Thus m [p] and so p = m. ⊆ ⊗ 6 ∈

Theorem 3.4. If the topology of a scheme X is Hausdorff, then every finite union of affine opens of X is an affine open. In particular, every compact scheme is an affine scheme.

Proof. Using the induction, then it suffices to prove the assertion for two cases. Hence, let U and V be two affine opens of X. Every affine open of X is closed, because in a Hausdorff space each quasi- compact subset is closed. It follows that W := U V is a clopen of U. Therefore W , U W and V W are affine opens,∩ see Theorem 2.2. It is well known that\ if a scheme\ can be written as the disjoint union of a finite number of affine opens, then it is an affine scheme. Hence, U V is an affine open. ∪

We use the above theorems to obtain more geometric results.

Corollary 3.5. The category of compact (aﬃne) schemes is anti-equivalent to the category of zero dimensional rings.

Corollary 3.6. Let X be a scheme which has an aﬃne open covering such that the intersection of any two elements of this covering is quasi- compact. Then the underlying space of X is Hausdorﬀ if and only if every point of X is a closed point.

Proof. The implication “ ” is obvious since each point of a Haus- dorff space is a closed point.⇒ Conversely, if X = Spec(A) is an affine scheme then every prime of A is a maximal ideal. Thus by Theorem 3.3 (ii), Spec(A) is Hausdorff. For the general case, let x and y be two distinct points of X. By the hypothesis, there exist affine opens U and V of X such that x U, y V and U V is quasi-compact. If either x V or y U then∈ the assertion∈ holds.∩ Because, by what we have ∈ ∈ 8 M.AGHAJANIANDA.TARIZADEH proved above, every affine open of X is Hausdorff. Therefore we may assume that x / V and y / U. But W := V (U V ) is an open subset of X because every∈ quasi-compact∈ (=compact)\ ∩ subset of a Hausdorff space is closed. Clearly y W and U W = . ∈ ∩ ∅

Remark 3.7. The hypothesis of Corollary 3.6 is not limitative at all. Because a separated scheme or more generally a quasi-separated scheme has this property, see [21, Proposition 3.6] or [19, Ex. 4.3] for the separated case and [14, Tag 054D] for the quasi-separated case.

4. Gelfand rings Lemma 4.1. Let S be a multiplicative subset of a ring A. Then the canonical morphism π : A S−1A is surjective if and only if Im π∗ = p Spec(A): p S = →is a Zariski closed subset of Spec(A). { ∈ ∩ ∅}

Proof. The map π∗ : Spec(S−1A) Spec(A) is a homeomorphism onto its image. If Im π∗ is Zariski→ closed, then π∗ is a closed −1 −1 map. If q Spec(S A), then the morphism Ap (S A)q induced by π is an∈ isomorphism where p = π−1(q). Therefore→ the morphism (π∗, π♯) : Spec(S−1A) Spec(A) is a closed immersion of schemes. It is well known that a morphism→ of rings A B is surjective if and only if the induced morphism Spec(B) Spec(→ A) is a closed immersion of schemes. Hence, π is surjective.→ Conversely, if π is surjective then Im π∗ = V (Ker π).

If p is a prime ideal of a ring A, then the image of each f A under the canonical ring map π : A A is also denoted by f . ∈ p → p p

Remark 4.2. Let A be a ring and consider the following relation R = (p, q) X2 : p + q = A on X = Spec(A). Clearly it is reflexive and symmetric.{ ∈ Let 6 be the} equivalence relation generated by R. ∼R Then p R q if and only if there exists a finite set p1, ..., pn of prime ideals of∼A with n > 2 such that p = p, p = q and{ p + p } = A for 1 n i i+1 6 all 1 6 i 6 n 1. Note that it may happen that p R q but p + q = A. Obviously Spec(− A)/ = [m]: m Max A = [p∼]: p Min A . It is ∼R { ∈ } { ∈ } important to notice that in Theorem 4.3 (xi) by Spec(A)/ R we mean the quotient of the “Zariski” space Spec(A) modulo .∼ By contrast, ∼R in Theorem 6.2 (viii), by Spec(A)/ R we mean the quotient of the “flat” space Spec(A) modulo . Also∼ note that in Theorems 4.3 and ∼R GELFAND RINGS, CLEAN RINGS AND THEIR DUAL RINGS 9

6.2, by [p] we mean the equivalence class of containing p. ∼R

Recall that a ring A is called a Gelfand ring if each prime ideal of A is contained in a unique maximal ideal of A. In the following result, the criteria (i)-(v) are well known, and the remaining are new.

Theorem 4.3. For a ring A the following statements are equivalent. (i) A is a Gelfand ring. (ii) Max(A) is the Zariski retraction of Spec(A). (iii) Spec(A) is a normal space with respect to the Zariski topology. (iv) If f A, then there exist g, h A such that (1+ fg)(1+ f ′h)=0 where f ′ =1∈ f. ∈ (v) If m and−m′ are distinct maximal ideals of A, then there exist f A m and g A m′ such that fg =0. ∈ \ ∈ \ (vi) If m is a maximal ideal of A, then the canonical map πm : A Am is surjective. → (vii) If m is a maximal ideal of A, then [m]=Λ(m). (viii) If m is a maximal ideal of A, then Λ(m) is a Zariski closed subset of Spec(A). ′ (ix) If m and m are distinct maximal ideals of A, then Ker πm + Ker πm′ = A. (x) If m and m′ are distinct maximal ideals of A, then there exists some f A such that f =0 and f ′ =1. ∈ m m (xi) The map η : Max(A) Spec(A)/ R given by m [m] is a homeomorphism. → ∼ (xii) If m is a maximal ideal of A, then Λ(m)= V (Ker πm).

Proof. For (i) (ii) (iii) see [15, Theorem 1.2], and for (i) (iv) (v) see⇔ [10, Theorem⇔ 4.1] and [11, Proposition 1.1]. ⇔ ⇔ (v) (vi) : It suﬃces to show that the morphism ϕm′ : Am′ (Am)m′ ⇒ ′ → induced by ϕ := πm is surjective for all m Max(A). Clearly ϕm is an ′ ∈ isomorphism. If m = m, then by Lemma 3.1, (Am)m′ Am A Am′ = 0. (vi) (v) : Choose6 some h m′ m then there exists≃ some⊗ a A ⇒ ∈ \ ∈ such that 1/h = a/1 in Am. Thus there exists some f A m such that f(ah 1) = 0. Clearly g := ah 1 A m′. ∈ \ (i) (vii)− : Let p [m]. There exists− a∈ maximal\ ideal m′ of A such ⇒ ′ ∈ ′ that p m . It follows that m R m . Thus there exists a ﬁnite set p , ..., ⊆p of prime ideals of A with∼ n > 2 such that p = m, p = m′ { 1 n} 1 n and pi + pi+1 = A for all 1 6 i 6 n 1. By induction on n we shall prove that m =6 m′. If n = 2 then m +−m′ = A and so m = m′. Assume 6 ′ that n > 2. We have p − + p − = A and p − m . Thus by the n 2 n 1 6 n 1 ⊆ 10 M.AGHAJANIANDA.TARIZADEH

′ ′ hypothesis, pn−2 m and so pn−2 + m = A. Thus in the equivalence ′ ⊆ 6 m R m the number of the involved primes is reduced to n 1. There- fore∼ by the induction hypothesis, m = m′. − (vii) (i) : Let p be a prime of A such that p m and p m′ for ⇒ ′ ⊆ ⊆′ some maximal ideals m and m of A. It follows that m R m and so [m] = [m′]. Thus by the hypothesis, m = m′. ∼ (vi) (viii) : See Lemma 4.1. ⇔ (vi) (ix) : We have V (Ker πm) = p Spec(A): p m . If ⇒ { ∈ ′′⊆ } Ker πm + Ker πm′ = A, then there exists a maximal ideal m such that 6 ′′ ′′ ′ Ker πm + Ker πm′ m . This yields that m = m = m which is a contradiction. ⊆ ′ ′ (ix) (v) : There are f Ker πm and g Ker πm′ such that f ′ + g⇒′ = 1. Thus there exist∈f A m and g∈ A m′ such that ff ′ = gg′ = 0. It follows that fg =∈ 0. \ ∈ \ (ix) (x) : There exists some f Ker πm such that 1 f Ker πm′ . (x) ⇒ (v) : There exist g A ∈m and h A m′ such− that∈ fg = 0 and⇒ (1 f)h = 0. It follows∈ that\ gh = 0. ∈ \ (i) (−xi) : For any ring A the map η is continuous and surjective, see⇒ Remark 4.2. By (vii), it is injective. It remains to show that its converse µ is continuous. By (ii), γ = µ π is continuous where ◦ π : Spec(A) Spec(A)/ R is the canonical morphism. Therefore µ is continuous.→ ∼ (xi) (i) : It is proved exactly like the implication (vii) (i). ⇒ ⇒ (ix) (xii) : Clearly Λ(m) V (Ker πm). Let p be a prime ideal of A ⇒ ⊆ ′ such that Ker πm p. If p * m, then there exists a maximal ideal m of A such that p ⊆m′. Using the hypothesis, then we get that m′ = A which is a contradiction.⊆ (xii) (viii) : There is nothing to prove. ⇒

Remark 4.4. Clearly Gelfand rings are stable under taking quotients, and mp-rings are stable under taking localizations. But the localization of a Gelfand ring is not necessarily a Gelfand ring. For example, consider the prime ideals p = (x/1) and q = (y/1) in A = (k[x, y])P where k is a domain and P = (x, y). Then A is a Gelfand ring but S−1A is not a Gelfand ring where S = A p q. Dually, k[x, y] is a mp-ring but the quotient A = k[x, y]/I is\ not∪ a mp-ring, because p =(x + I) and q =(y + I) are two distinct minimal primes of A which are contained in the prime ideal (x + I,y + I) where I =(xy).

Remark 4.5. Note that the main result of [24, Theorem 2.11 (iv)] by Harold Simmons is not true and the gap is not repairable. It claims that GELFAND RINGS, CLEAN RINGS AND THEIR DUAL RINGS 11 if Max(A) is Zariski Hausdorﬀ, then A is a Gelfand ring. As a coun- terexample for the claim, let p and q be two distinct prime numbers, S = Z (pZ qZ) and A = S−1Z. Then Max(A)= S−1(pZ),S−1(qZ) \ ∪ −{ } is Zariski Hausdorﬀ because D(p/1) Max A = S 1(qZ) and D(q/1) − ∩ { } ∩ Max(A)= S 1(pZ) . But A is not a Gelfand ring, since it is nonlocal domain. See{ also [25],} Simmons’ erratum for [24]. In Proposition 4.6, we correct his mistake.

Proposition 4.6. If Max(A) is Zariski Hausdorﬀ and N = J, then A is a Gelfand ring where J is the Jacobson radical of A.

Proof. Let m and m′ be distinct maximal ideals of A both containing a prime p of A. By the hypotheses, there are f A m and g A m′ such that D(f) Max(A) D(g) Max(∈A) \= . It ∈ \ ∩ ∩ ∩ ∅ follows that fg J. Thus there exists a natural number n> 1 such that f ngn = 0.∈ Then either f p or g p. This is a contradiction, hence A is a Gelfand ring. ∈ ∈

Corollary 4.7. Let A be a ring. Then Max(A) is Zariski Hausdorﬀ if and only if A/J is a Gelfand ring.

5. Clean rings In this section we give new characterizations for clean rings. By a system of equations over a ring A we mean a ﬁnite number of equations f (x , ..., x ) = 0 with i =1, ..., d where each f (x , ..., x ) i 1 n i 1 n ∈ A[x1, ..., xn]. We say that this system has a solution in A if there exists n an n-tuple (c1, ..., cn) A such that fi(c1, ..., cn) = 0 for all i. Remember that a ring∈ A is called a clean ring if each f A can be written as f = e + u where e A is an idempotent and∈ u A is invertible in A. In the following∈ result, the criteria (i)-(vii) are∈ well known, and the remaining are new.

Theorem 5.1. For a ring A the following statements are equivalent. (i) A is a clean ring. (ii) A is an exchange ring, i.e., for each f A there exists an idempotent e A such that e Af and 1 e A∈(1 f). (iii) A∈is a Gelfand ring∈ and every− pure∈ ideal− of A is a regular ideal. (iv) The collection of D(e) Max(A) with e A an idempotent forms a basis for the induced Zariski∩ topology on Max(∈ A). 12 M.AGHAJANIANDA.TARIZADEH

(v) The idempotents of A can be lifted modulo each ideal of A. (vi) A is a Gelfand ring and Max(A) is totally disconnected with respect to the Zariski topology. (vii) A is a Gelfand ring and for each maximal ideal m of A, Ker πm is a regular ideal of A. (viii) If a system of equations over an A algebra B has a solution in − each ring Bm with m a maximal ideal of A, then that system has a solution in the ring B. (ix) If m and m′ are distinct maximal ideals of A, then there exists an idempotent e A such that e m and 1 e m′. (x) If m and ∈m′ are distinct maximal∈ ideals− of∈ A, then there exists an idempotent e A such that e Ker πm and 1 e Ker πm′ . (xi) The connected∈ components∈ of Spec(A) are− precisely∈ of the form Λ(m) where m is a maximal ideal of A. (xii) The map λ : Max(A) Sp(A) given by m (f m : f = f 2) is a homeomorphism. → ∈

Proof. In order to see the equivalences of (i) (v) consider [22, Theorem 1.7]), and for the equivalences of (i), (vi−) and (vii) see [13, Theorem 1.1]. (vi) (viii) : Consider the system of equations fi(x1, ..., xn) = 0 with i = 1⇒, ..., d where f (x , ..., x ) B[x , ..., x ]. (If ϕ : A B is the i 1 n ∈ 1 n → structure morphism then as usual a.1B = ϕ(a) is simply denoted by a for all a A). Using the calculus of fractions, then we may ﬁnd a ∈ positive integer N and polynomials gi(y1, ..., yn, z1, ..., zn) over B such that N fi(b1/s1, ..., bn/sn)= gi(b1, ..., bn,s1, ..., sn)/(s1...sn) for all i and for every b1, ..., bn B and s1, ..., sn S where S is a multiplicative subset of A. If the∈ above system has∈ a solution in each ring Bm, then there exist b1, ..., bn B and c,s1, ..., sn A m such that ∈ ∈ \

cgi(b1, ..., bn,s1, ..., sn)=0 for all i. This leads us to consider C the collection of those opens W of Max(A) such that there exists b1, ..., bn B and c,s1, ..., sn A ( m) so that ∈ ∈ \ ∈ mSW cgi(b1, ..., bn,s1, ..., sn)=0. Clearly C covers Max(A) and if W C then every open subset of W is also a member of C . Thus by Theorem∈ 2.3, we may ﬁnd a ﬁnite number W , ..., W C of pairwise disjoint clopens of Max(A) such 1 q ∈ GELFAND RINGS, CLEAN RINGS AND THEIR DUAL RINGS 13

q that Max(A) = Wk. Using the retraction map γ : Spec(A) k=1 → Max(A) and TheoremS 2.2, then the map f D(f) Max(A) is a bijection from the set of idempotents of A onto the set∩ of clopens of Max(A). Therefore there exist orthogonal idempotents e1, ..., eq A q ∈ such that Wk = D(ek) Max(A). Clearly ek is an idempotent and ∩ k=1 q q P D( ek) = Spec(A). It follows that ek = 1. For each k = 1, ..., q kP=1 kP=1 there exist b1k, ..., bnk B and ck,s1k, ..., snk A ( m) such that ∈ ∈ \ ∈ m SWk ckgi(b1k, ..., bnk,s1k, ..., snk)=0 q ′ ′ for all i. For each j = 1, ..., n setting bj := ekbjk and sj := k=1 q P eksjk. Note that for each natural number p > 0 we have then k=1 P q q ′ p p ′ p p (bj) = ek(bjk) and (sj) = ek(sjk) . We claim that kP=1 kP=1 ′ ′ ′ ′ ′ c gi(b1, ..., bn,s1, ..., sn)=0 q ′ for all i where c := ekck. Because ﬁx i and let kP=1 i1 in in+1 i2n gi(y1, ..., yn, z1, ..., zn)= ri1,...,i2n y1 ...yn z1 ...zn . 06i1,....,iX2n<∞ Then ′ ′ ′ ′ ′ c gi(b1, ..., bn,s1, ..., sn)= q ′ i1 in in+1 i2n c ri1,...,i2n ek(b1k) ...(bnk) (s1k) ...(snk) = 06i1,....,iX2n<∞ Xk=1 q q

( etct) ekgi(b1k, ..., bnk,s1k, ..., snk) = Xt=1 Xk=1 q

ekckgi(b1k, ..., bnk,s1k, ..., snk)=0. Xk=1 This establishes the claim. But c′ is invertible in A since c′ / m for ′ ′ ′ ′ ∈ all m Max(A). Hence, gi(b1, ..., bn,s1, ..., sn) = 0 for all i. Similarly, ∈′ each sj is invertible in A. Therefore ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ N fi(b1/s1, ..., bn/sn)= gi(b1, ..., bn,s1, ..., sn)/(s1...sn) =0 14 M.AGHAJANIANDA.TARIZADEH

′′ ′′ n for all i. Hence, the n-tuple (b1, ..., bn) B is a solution of the above ′′ ′ ′ −1 ∈ system where bj := bjϕ(sj) . (viii) (ii) : It suffices to show that the system of equations: ⇒ X = X2  X = fY  1 X = (1 f)Z − −  has a solution in A. If A is a local ring with the maximal ideal m, then the system having the solution 0, 0, 1/(1 f) or (1, 1/f, 0), according − as f m or f / m. Using this, then by the hypothesis the system has a solution∈ for every∈ ring A (not necessarily local). (vii) (ix) : If m and m′ are distinct maximal ideals of A, then by ⇒ Theorem 4.3, Ker πm + Ker πm′ = A. Thus there exists an idempotent ′ ′ e Ker πm m such that e / m . It follows that 1 e m . (ix∈) (vi)⊆ : It is clear. ∈ − ∈ (ix) ⇒ (x) : Easy. ⇔ (vi) (xi) : If m is a maximal ideal of A, then A/ Ker πm has no nontrivial⇒ idempotents since by Theorem 4.3 (vi), it is canonically iso- morphic to Am. By (vii), Ker πm is a regular ideal. It follows that Ker πm is a max-regular ideal of A. Hence, by [26, Theorem 3.17], V (Ker πm) is a connected component of Spec(A). Conversely, if C is a connected component of Spec(A), then γ(C) is a connected subset of Max(A) where γ : Spec(A) Max(A) is the retraction map which sends each prime ideal p of A→into the unique maximal ideal of A containing p. Therefore there exists a maximal ideal m of A such that γ(C) = m , because Max(A) is totally disconnected. We have −1 { } C γ ( m )=Λ(m)= V (Ker πm). It follows that C = V (Ker πm). (xi⊆) (vi{ )} : Clearly A is a Gelfand ring, because distinct connected components⇒ are disjoint. The map ϕ : Max(A) π Spec(A) given → 0 by m Λ(m) is bijective. It is also continuous and closed map, because ϕ = π i and π Spec(A) is Hausdorff where i : Max(A) Spec(A) is ◦ 0 → the canonical injection and π : Spec(A) π Spec(A) is the canoni- → 0 cal projection. Therefore, Max(A) is totally disconnected. (vii) (xii) : The map λ is continuous. It is also a closed map, because⇒ for any ring A, Max(A) is quasi-compact and Sp(A) is Hausdorff. If m is a maximal ideal of A, then by the hypothesis, λ(m) = Ker πm. This yields that λ is injective, because A is a Gelfand ring. If M is a max-regular ideal of A, then there exists a maximal ideal m of A such that M m. It follows that M Ker πm = λ(m). This yields that M = λ(m⊆). Hence, λ is surjective.⊆ (xii) (vi) : By the hypothesis, Max(A) is totally disconnected and ⇒ GELFAND RINGS, CLEAN RINGS AND THEIR DUAL RINGS 15 it is the retraction of Spec(A). Hence, A is a Gelfand ring.

Corollary 5.2. The max-regular ideals of a clean ring A are precisely of the form Ker πm where m is a maximal ideal of A.

Proof. If M is a max-regular ideal of A, then V (M) is a connected component of Spec(A). Thus by Theorem 5.1 (xi), there exists a maximal ideal m of A such that V (M) = Λ(m). It follows that M m. This yields that M Ker π . But Ker π is a regular ideal of A⊆, see ⊆ m m Theorem 5.1 (vii). Therefore M = Ker πm. By a similar argument, it is proven that if m is a maximal ideal of A, then Ker πm is a max-regular ideal of A.

Corollary 5.3. [4, Theorem 9] If A/N is a clean ring, then A is a clean ring.

Proof. It implies from Theorem 5.1 (xi).

Corollary 5.4. [4, Corollary 11] Every zero dimensional ring is a clean ring.

Proof. If A is a zero dimensional ring then by Theorem 3.3 (iii), the Zariski and patch topologies over Max(A) are the same and so it is totally disconnected. Therefore by Theorem 5.1 (vi), A is a clean ring.

6. Mp-rings In this section mp-rings are characterized.

Remark 6.1. We observed that if A is a Gelfand ring, then Max(A) is Zariski Hausdorff. Dually, if A is a mp-ring then Min(A) is flat Hausdorff, because if p and q are distinct minimal primes of A then p + q = A, thus there are f p and g q such that f + g = 1 and so V (f) V (g)= . ∈ ∈ ∩ ∅

The following result is the culmination of mp-rings. 16 M.AGHAJANIANDA.TARIZADEH

Theorem 6.2. For a ring A the following statements are equivalent. (i) A is a mp-ring. (ii) If p and q are distinct minimal prime ideals of A, then p + q = A. (iii) A/N is a mp-ring. (iv) If p is a minimal prime ideal of A, then [p]= V (p). (v) Min(A) is the flat retraction of Spec(A). (vi) Spec(A) is a normal space with respect to the flat topology. (vii) If p is a minimal prime ideal of A, then V (p) is a flat closed subset of Spec(A). (viii) The map η : Min(A) Spec(A)/ R given by p [p] is a homeomorphism. → ∼ (ix) If fg = 0 for some f,g A, then there exists a natural number n > 1 such that Ann(f n) + Ann(∈ gn)= A. (x) Every minimal prime ideal of A is the radical of a unique pure ideal of A.

Proof. The implications (i) (ii) (iii) (i) are easy. (i) (iv) : Let q [p]. There⇒ exists⇒ a minimal⇒ prime ideal p′ of A such⇒ that p′ q. It∈ follows that p p′. Then by Remark 4.2, there ⊆ ∼R exists a finite set q1, ..., qn of prime ideals of A with n > 2 such that ′ { } q1 = p, qn = p and qi + qi+1 = A for all 1 6 i 6 n 1. By induction on n we shall prove that p =6 p′. If n = 2 then p +−p′ = A and so by the hypothesis, p = p′. Assume that n > 2. There exists6 a minimal ′′ ′′ ′ prime ideal p of A such that p qn−1. We have qn−1 + p = A. Thus ′ ′′ ⊆ ′ 6 by the hypothesis, p = p . It follows that qn−2 + p = A. Thus in the ′ 6 equivalence p R p the number of the involved primes is reduced to n 1. Therefore∼ by the induction hypothesis, p = p′. (iv−) (i) : Let q be a prime ideal of A such that p q and p′ q for ⇒ ′ ⊆ ′ ⊆ some minimal primes p and p of A. This implies that p R p and so [p] = [p′]. This yields that p = p′. ∼ (i) (v) : Consider the function γ : Spec(A) Min(A) which sends each⇒ prime ideal p of A into the unique minimal→ prime ideal of A which is contained in p. It suffices to show that γ−1 D(f) Min(A) is a ∩ flat closed subset of Spec(A) for all f A. LetE := D(f) Min( A). We show that γ−1(E) = Im π∗ where ∈π : A S−1A is the∩ canonical ring map and S = A p. Clearly γ−1(→E) Im π∗. Conversely, \ ⊆ f∈ /Sγ(p) if q Im π∗ then q p. If f γ(q), then for each p E there ∈ ⊆ ∈ ∈ f∈ /Sγ(p) exist x p and y γ(q) such that x + y = 1. It follows that p ∈ p ∈ p p E V (xp). But E is a flat closed subset of Min(A) and for any ⊆ ∈ pSE GELFAND RINGS, CLEAN RINGS AND THEIR DUAL RINGS 17 ring A the subspace Min(A) is flat quasi-compact. This yields that E is n n flat quasi-compact. Hence E V (x )= V (x) where x = x and ⊆ i i iS=1 iQ=1 xi := xpi for all i. Thus we may find some y γ(q) such that x+y = 1. Hence there exists a prime ideal p such that∈f / γ(p) and y p. This yields that 1 = x + y p, a contradiction. Therefore∈ f / γ(q∈). (v) (i) : Let p be∈ a minimal prime ideal of A such∈ that p q for some⇒ prime ideal q of A. By the hypothesis there exists a retrac-⊆ tion map ϕ : Spec(A) Min(A). Clearly p Λ(q) = q and so → ∈ { } p = ϕ(p) ϕ(q) = Λ ϕ(q) Min(A) = ϕ(q) . It follows that ∈ { } ∩ { } p = ϕ(q). (i) (vi) : If E and F are disjoint flat closed subsets of Spec(A), then⇒γ(E) γ(F )= , because flat closed subsets are stable under the generalization∩ where∅γ : Spec(A) Min(A) is the retraction map. By (v), the map γ is continuous.→ The space Min(A) is also flat Haus- dorff, see Remark 6.1. Therefore γ is a closed map. But Min(A) is a normal space, because it is well known that every compact space is a normal space. Thus there exist opens γ(E) U and γ(F ) V in Min(A) such that U V = . It follows that⊆ E γ−1(U)⊆ and F γ−1(V ) are disjoint∩ opens of∅ Spec(A). ⊆ (vi⊆) (ii) : If p and q are distinct minimal prime ideals of A, then by the⇒ hypothesis there exists (finitely generated) ideals I and J of A such that p V (I), q V (J) and V (I) V (J) = . This yields that p + q = A. ∈ ∈ ∩ ∅ (i) (vii) : By [26, Theorem 3.11], it suffices to show that V (p) is stable⇒ under the generalization. Let q′ q be prime ideals of A such that p q. There exists a minimal prime⊆ ideal p′ of A such that p′ q′. This⊆ yields that p = p′, since A is a mp-ring. ⊆ (vii) (ii) : If p + q = A, then there exists a maximal ideal m of A such⇒ that p + q m6. It follows that m V (p). This yields that q V (p), because by⊆ the hypothesis, V (p) is∈ stable under the generalization.∈ This implies that p = q. But this is a contradiction. (i) (viii) : This is proved exactly like the proof of the implication (i) ⇒(xi) of Theorem 4.3. (viii⇒) (i) : It is proved exactly like the implication (iv) (i). (i) ⇒(ix) : Suppose Ann(f n) + Ann(gn) = A for all n⇒> 1. Then I :=⇒ Ann(f n) + Ann(gn) is a proper ideal6 of A. So there exists a n>1 maximalP ideal m of A such that I m. Let p be the unique minimal ⊆ prime ideal of A contained in m. If N is the nil-radical of Am, then n Clearly f/1 / N = pAm, because Ann(f ) m for all n > 1. There- fore f / p.∈ By a similar argument, we get⊆ that g / p. But this is a ∈ ∈ 18 M.AGHAJANIANDA.TARIZADEH contradiction, since fg = 0. (ix) (ii) : By Lemma 3.2, there exist f A p and g A q such that ⇒fg = 0. Thus by the hypothesis, Ann(∈ f\n) + Ann(∈gn) =\ A for some n > 1. Hence, there exist x Ann(f n) and y Ann(gn) such that 1 = x + y. It follows that x p∈and y q and so∈p + q = A. (vii) (x) : By [26, Theorem 3.11],∈ V (p) is∈ stable under the generalization.⇒ Thus by [27, Theorem 3.2], there exists a unique pure ideal I of A such that V (p)= V (I). So p = √I. (x) (ii) : There exists some f p such that f / q. By the hypothe- ⇒ ∈ ∈ sis, there exists a pure ideal I of A such that p = √I. So by Theorem 2.1, there exist some g I and a natural number n > 1 such that f n(1 g) = 0. It follows∈ that 1 g q. Thus p + q = A. − − ∈ If p is a minimal prime ideal of a mp-ring A, then the unique pure ideal in Theorem 6.2 (x) associated to p, is the largest pure ideal of A which is contained in Ker πp. We give a new proof to the following technical result. Our proof essen- tially differs with [13, §4] from the scratch.

Theorem 6.3. Let A be a mp-ring. Then the retraction map γ : Spec(A) Min(A) is Zariski continuous if and only if Min(A) is Zariski compact.→

Proof. If γ is Zariski continuous, then clearly Min(A) is Zariski quasi-compact. For any ring A, then Min(A) is Zariski Hausdorff, see Lemma 3.2. Thus Min(A) is Zariski compact. To see the converse it suffices to show that γ−1(U) is a Zariski open of Spec(A) where U = Min(A) D(f) and f A. It is well known and easy to see that for any ring A, then∩ U is a Zarisiki∈ clopen of Min(A), see [28, Proposition 4.2]. Thus there exists an ideal I of A such that U = Min(A) V (I)= Min(A) V (g). It follows that U c = Min(A) U = Min(A∩) D(g). ∈ ∩ \ ∈ ∩ gTI gSI By the hypothesis, Min(A) is Zariski quasi-compact. Thus there exist a n c finitely many elements g1, ..., gn I such that U = Min(A) D(gi), ∈ i=1 ∩ since every closed subset of a quasi-compact spaceS is quasi-compact. This yields that U = Min(A) V (J) where J = (g1, ..., gn). Thus U is a flat open of Min(A). It∩ is also a flat closed subset of Min(A). Therefore γ−1(U) is a flat clopen of Spec(A), because by Theorem 6.2 (v), the map γ is flat continuous. But Zariski clopens and flat clopens of Spec(A) are the same, see [26, Corollary 3.12]. Hence, γ−1(U) is a GELFAND RINGS, CLEAN RINGS AND THEIR DUAL RINGS 19

Zariski open.

If A is a ring, then clearly Ann(f) + Ann(g) Ann(fg) for all f,g A. In Theorem 6.4, it is shown that the equality⊆ holds if and only∈ if A is a reduced mp-ring. In Theorem 6.4, we also improve [2, Theorem 2.6], [6, Lemma β ] and [7, Theorem 2.3] by adding the con- dition (viii) as a new equivalent.

Theorem 6.4. For a ring A the following statements are equivalent. (i) A is a reduced mp-ring. (ii) If fg =0 for some f,g A, then Ann(f) + Ann(g)= A. (iii) Ann(f) + Ann(g) = Ann(∈ fg) for all f,g A. (iv) Ann(f) is a pure ideal for all f A. ∈ (v) Every principal ideal of A is a ﬂat∈ A module. − (vi) If p is a prime ideal of A, then Ap is an integral domain. (vii) If m is a maximal ideal of A, then Am is an integral domain. (viii) Every minimal prime ideal of A is a pure ideal.

Proof. (i) (ii) : By Theorem 6.2 (ix), Ann(f n) + Ann(gn) = A for some n > ⇒1. But Ann(f) = Ann(f k) for all f A and all k > 1, since A is reduced. ∈ (ii) (i) : By Theorem 6.2 (ix), A is a mp-ring. Let f be a nilpotent element⇒ of A. Thus there exists the least positive natural number n such that f n = 0. We show that n =1. If n> 1 then by the hypothesis, Ann(f n−1) = Ann(f) + Ann(f n−1) = A. It follows that f n−1 = 0. But this is in contradiction with the minimality of n. (ii) (iii) : If a Ann(fg) then (af)g = 0. Thus by the hypothesis,⇒ Ann(af) +∈ Ann(g) = A. Hence there are b Ann(af) and c Ann(g) such that b + c = 1. We have a = ab + ac∈, ab Ann(f) and∈ ac Ann(g). Thus a Ann(f) + Ann(g). The implications∈ (iii) (∈ii), (ii) (iv) (v)∈ and (vi) (vii) (ii) are easy. (i) ⇒(vi) : If q⇔is the minimal⇔ prime ideal⇒ of A⇒contained in p, then ⇒ qAp = 0. Hence, Ap is a domain. (i) (viii) : Let p be a minimal prime ideal of A and f p. If Ann(⇒f)+ p = A, then there exists a maximal ideal m of A such∈ that 6 Ann(f)+ p m. By the hypotheses, pAm = 0. Hence there exists some g A ⊆m such that fg = 0. But this is a contradiction. Thus by Theorem∈ 2.1,\ p is a pure ideal. (viii) (i) : By Theorem 6.2 (x), A is a mp-ring. If f A is nilpotent, then Ann(⇒ f) = A. If not, then there exists a maximal∈ ideal m of A such that Ann(f) m. There exists a minimal prime ideal p of A such ⊆ 20 M.AGHAJANIANDA.TARIZADEH that p m. Clearly f p and so by the hypothesis, there is some g p such⊆ that 1 g Ann(∈ f) which is a contradiction. Thus f = 0. Hence,∈ A is reduced.− ∈

In the literature, a ring A is said to be a p.f. ring if every principal ideal of A is a ﬂat A module. By Theorem 6.4, p.f. rings and reduced mp-rings are the same.−

Corollary 6.5. If p is a prime ideal of a reduced mp-ring A, then Ker πp is the unique minimal prime ideal of A which is contained in p.

Proof. Clearly Ker πp p, and by Theorem 6.4 (ii), it is a prime ideal. Its minimality is easily⊆ veriﬁed and uniqueness is obvious, since A is a mp-ring.

7. Pure ideals of reduced Gelfand rings and mp-rings In this section, new characterizations for pure ideals of reduced Gelfand rings and mp-rings are given. The following result generalizes [1, Theorem 1.8] to any ring.

Lemma 7.1. If I is a pure ideal of a ring A, then:

I = Ker πm. m∈Max(\A)∩V (I)

Proof. If m is a maximal ideal of A containing I, then I Ker πm, because if f I then by Theorem 2.1, there exists some g⊆ I such that (1 g)f ∈= 0, clearly 1 g A m and so f Ker π . Conversely,∈ − − ∈ \ ∈ m if f Ker πm then to prove f I it suﬃces to show that ∈ ∈ ∩ ∈ m Max(TA) V (I) Ann(f)+ I = A. If Ann(f)+ I = A, then there exists a maximal ′ 6 ′ ideal m of A such that Ann(f)+ I m and so f / Ker πm′ which is a contradiction. ⊆ ∈

Theorem 7.2. The pure ideals of a reduced Gelfand ring A are precisely of the form Ker πm where E is a Zariski closed subset m∈Max(TA)∩E of Spec(A). GELFAND RINGS, CLEAN RINGS AND THEIR DUAL RINGS 21

Proof. It follows from Lemma 7.1 and [1, Theorem 1.9].

Theorem 7.3. The pure ideals of a reduced mp-ring A are precisely of the form p where E is a ﬂat closed subset of Spec(A). p∈Min(TA)∩E

Proof. If I is a pure ideal of A, then V (I) is a flat closed subset of Spec(A) and we have I = p. Because if we choose p∈Min(TA)∩V (I) f in the intersection, then to prove f I it suffices to show that Ann(f)+ I = A. If Ann(f)+ I = A, then∈ Ann(f)+ I m for some maximal ideal m of A. There exists6 a minimal prime ideal⊆ q of A such that q m. But I q, since I is a pure ideal and so by Theorem 2.1, V (I) is⊆ stable under⊆ the generalization. Hence, f q. By Theorem 6.4, q is a pure ideal. Thus by Theorem 2.1, there∈ exists some g q such that (1 g)f = 0. It follows that 1 = (1 g)+ g m which∈ is a contradiction.− Conversely, setting J := − p where∈ E is a flat p∈Min(TA)∩E closed subset of Spec(A). Let f J. Then for each p Min(A) E there exists some c p such that∈ f = fc , because by∈ Theorem∩ 6.4, p ∈ p p is a pure ideal of A. We claim that J + (1 cp : p Min(A) E) is the unit ideal of A. If not, then it is contained− in a∈ maximal∩ ideal m of A. There exists a minimal prime ideal q of A such that q m. ⊆ Clearly q / E. Thus for each p Min(A) E there exist xp p and y q∈such that x + y = 1,∈ because A∩is a mp-ring. We have∈ p ∈ p p E V (xp), because every flat closed subset of Spec(A) is ⊆ ∈ ∩ p Min(SA) E stable under the generalization. But every closed subset of a quasi- compact space is quasi-compact. Therefore there exist a finite number n p1, ..., pn Min(A) E such that E V (xi) where xi := xpi for ∈ ∩ ⊆ i=1 S n all i. There exists some y q such that x + y = 1 where x = xi. ∈ i=1 Clearly x J. It follows that 1 = x + y J + q m. ThisQ is a contradiction.∈ Therefore J + (1 c : p Min(∈ A) E⊆)= A. Thus we − p ∈ ∩ may write 1 = g + ak(1 ck) where g J and ck := cpk for all k. It k − ∈ follows that f = fgP. Therefore by Theorem 2.1, J is a pure ideal.

Corollary 7.4. [2, Theorems 2.4 and 2.5] The pure ideals of a reduced mp-ring A are precisely of the form Ker πm where I is an m∈Max(TA)∩V (I) 22 M.AGHAJANIANDA.TARIZADEH ideal of A.

Proof. The implication “ ” implies from Lemma 7.1. To prove the ⇒ converse, ﬁrst we claim that Ker πm : m Max(A) V (I) = p : p Min(A) E where E = Im{ π∗ and π : ∈A S−1A ∩is the canonical} { ring∈ map with∩ S}=1+ I. The inclusion implies→ from Corollary 6.5. To see the reverse inclusion, if p Min(A⊆) E then I + p = A. Thus there exists a maximal ideal m of∈ A such∩ that I + p m6. Then by ⊆ Corollary 6.5, p = Ker πm. Hence the claim is established. This yields that Ker πm = p. Now by Theorem 7.3, the as- m∈Max(TA)∩V (I) p∈Min(TA)∩E sertion is concluded.

Corollary 7.5. If A is a reduced mp-ring, then Ann(f) J = 0 for ∩ f all f A where Jf := Ker πm. ∈ ∈ ∩ m Max(TA) V (f) Proof. By Theorem 6.4 and Corollary 7.4, Ann(f) J is a pure ideal ∩ f and it is contained in the Jacobson radical of A. Hence, Ann(f) Jf = 0. ∩

The above corollary together with [2, Lemma 3.4] provide a short and straight proof for [2, Theorem 3.5].

Corollary 7.6. If p is a minimal prime ideal of a reduced mp-ring A, then p = Jf . ∈ fPp Proof. If f p then there exists some h p such that f(1 h) = 0. ∈ ∈ − This yields that f Jh. The reverse inclusion is deduced from Corol- lary 7.5. ∈

8. Purified rings The following deﬁnition introduces a new class of commutative rings.

Deﬁnition 8.1. We call a ring A a puriﬁed ring if for all distinct minimal prime ideals p and q there exists an idempotent in A which lies in p, but not in q. GELFAND RINGS, CLEAN RINGS AND THEIR DUAL RINGS 23

Obviously every purified ring is a mp-ring. The above definition, in the light of Theorem 5.1 (ix), is the dual notion of clean ring. It is important to notice that, in order to get the dual notion of clean ring, the initial definition of clean ring can not be dualized by replac- ing “product” instead of “sum”. Because each element of a ring can be written as a “product” of an invertible and an idempotent elements of that ring if and only if it is absolutely flat (von-Neumann regular) ring, see [4, page 1]. This in particular yields that every absolutely flat ring is again a clean ring, see Corollary 5.4 or [4, Theorem 10]. Purified rings are stable under taking localizations. A finite product of rings is a purified ring if and only if each factor is a purified ring.

Proposition 8.2. Every zero dimensional ring is a puriﬁed ring.

Proof. It follows from Theorem 5.1 (ix) and Corollary 5.4.

Theorem 8.3. Let A be a ring. Then A is a puriﬁed ring if and only if A/N is a puriﬁed ring.

Proof. Let A/N be a puriﬁed ring and let p and q be distinct minimal prime ideals of A. Then there exists an idempotent f + N A/N such that f p and 1 f q. Using Theorem 2.2, then it∈ is not hard to see that∈ the idempotents− ∈ of a ring A can be lifted modulo its nil-radical. So there exists an idempotent e A such that f e N. It follows that e p and 1 e = (1 f)+(f∈ e) q. − ∈ ∈ − − − ∈

Lemma 8.4. Let I be a regular ideal of a ring A. If f I, then there exists an idempotent e I such that f = fe. ∈ ∈

Proof. There exist a ﬁnitely many idempotents e , ..., e I such 1 n ∈ that f (e1, ..., en). But e1 + e2 e1e2 is an idempotent and (e1, e2)= (e + e∈ e e ). Thus by the induction,− there exists an idempotent 1 2 − 1 2 e I such that (e1, ..., en) = Ae. Therefore f = re for some r A. But∈ we have f(1 e)= re(1 e)=0 and so f = fe. ∈ − − The following result is the culmination of puriﬁed rings.

Theorem 8.5. For a reduced ring A the following statements are equivalent. 24 M.AGHAJANIANDA.TARIZADEH

(i) A is a purified ring. (ii) A is a mp-ring and Min(A) is totally disconnected with respect to the flat topology. (iii) Every minimal prime ideal of A is a regular ideal. (iv) The connected components of Spec(A) are precisely of the form V (p) where p is a minimal prime ideal of A. (v) If a system of equations over A has a solution in each ring A/p with p a minimal prime of A, then that system has a solution in A. (vi) The idempotents of A can be lifted along each localization of A. (vii) The collection of V (e) Min(A) with e A an idempotent forms a basis for the induced flat topology∩ on Min(A∈). (viii) A is a mp-ring and every pure ideal of A is a regular ideal. (ix) The max-regular ideals of A are precisely the minimal prime ideals of A. (x) For each f A, Ann(f) is a regular ideal of A. (xi) If fg =0 for∈ some f,g A, then there exists an idempotent e A such that f = fe and g = g(1∈ e). ∈ −

Proof. (i) (ii):If p and q are distinct minimal primes of A, then there exists an⇒ idempotent e A such that p V (e) and q V (1 e). We also have V (e) V (1 e∈) = Spec(A). Therefore∈ Min(A∈) is totally− disconnected with respect∪ − to the flat topology. (ii) (iii) : Let p be a minimal prime of A and f p. By Remark 6.1,⇒ Min(A) is flat Hausdorff. It is also flat quasi-compact.∈ There- fore by Theorem 2.3, there exists a clopen U Min(A) such that p U V (f) Min(A). Then by Theorem 2.2, there⊆ exists an idempotent∈ e ⊆ A such∩ that p V (e)= γ−1(U) where γ : Spec(A) Min(A) is the retraction∈ map, see∈ Theorem 6.2. We have γ−1(U) V→(f). Thus there exist a natural number n > 1 and an element a ⊆A such that f n = ae. It follows that 1 e Ann(f n). But Ann(f∈n) = Ann(f), since A is reduced. Therefore− f ∈= fe. (iii) (i) : Let p and q be distinct minimal primes of A. Then there exists⇒ an idempotent e p such that e / q. It follows that 1 e q. (ii) (iv) : If p is a∈ minimal prime∈ of A, then it is a max-regular− ∈ ideal⇒ of A, see the implication (ii) (iii). Thus by [26, Theorem 3.17], V (p) is a connected component of⇒A. Conversely, if C is a connected component of Spec(A), then there exists a minimal prime p of A such that γ(C) = p . But we have C γ−1( p ) = V (p). It follows that C = V (p). { } ⊆ { } (iv) (ii) : Clearly A is a mp-ring, because distinct connected components⇒ are disjoint. The map Min(A) π Spec(A) given by → 0 GELFAND RINGS, CLEAN RINGS AND THEIR DUAL RINGS 25 p V (p) is a homeomorphism. Hence, Min(A) is totally disconnected with respect to the flat topology. (ii) (v) : Assume that the system of equations fi(x1, ..., xn)=0 over⇒A has a solution in each ring A/p. Thus for each minimal prime ideal p of A there exist b1, ..., bn A such that fi(b1, ..., bn) p for all i. This leads us to consider C ,∈ the collection of those opens∈ W of Min(A) such that there exist b1, ..., bn A so that fi(b1, ..., bn) p ∈ ∈ ∈ pTW for all i. Clearly C covers Min(A), and if W C then every open subset of W is also a member of C . Thus by Theorem∈ 2.3, we may find a finite number W1, ..., Wq C of pairwise disjoint clopens of q ∈ Min(A) such that Min(A) = Wk. Using Theorem 2.2 and the re- k=1 traction map γ : Spec(A) SMin(A) of Theorem 6.2, then the map f V (f) Min(A) is a→ bijection from the set of idempotents of A onto the∩ set of clopens of Min(A). Therefore there are orthogonal idempotents e1, ..., eq A such that Wk = V (1 ek) Min(A). q ∈ q − ∩ Thus ek is an idempotent and D( ek) = Spec(A). It follows k=1 k=1 Pq P that e = 1. For each k = 1, ..., q there exist b , ..., b A k 1k nk ∈ kP=1 such that fi(b1k, ..., bnk) p for all i. For each j = 1, ..., n set- ∈ p∈Wk q T ′ ting bj := ekbjk. Note that if p > 0 is a natural number, then k=1 q P q ′ p p ′ ′ (bj) = ek(bjk) . It follows that fi(b1, ..., bn) = ekfi(b1k, ..., bnk) kP=1 kP=1 for all i. Now if p is a minimal prime of A then p Wt for some ′ ′ ∈ t. We have etfi(b1, ..., bn) = etfi(b1t, ..., bnt) p. This implies that ′ ′ ′ ′ ∈ fi(b1, ..., bn) p. Therefore fi(b1, ..., bn) p = 0 for all i. ∈ ∈ ∈ p Min(T A) (v) (vi) : Let S be a multiplicative subset of A. If a/s S−1A is an idempotent,⇒ then there exists some t S such that ast(a∈ s) = 0. It suffices to show that the following system:∈ −

X = X2 st(a sX)=0 − has a solution in A. If A is an integral domain, then the above system having the solution 0A or 1A, according as ast = 0 or a = s. Using this, then by the hypothesis the above system has a solution for every ring A (not necessarily domain). (vi) (i) : If p and q are distinct minimal prime ideals of A, then ⇒ 26 M.AGHAJANIANDA.TARIZADEH by the prime avoidance lemma, Spec(S−1A) = S−1p,S−1q where S = A (p q). Clearly S−1p and S−1q are distinct{ maximal ideals} of S−1A.\ Therefore∪ by Theorem 2.2, there exists an idempotent f S−1A such that D(f)= S−1q and D(1 f)= S−1p . By the hypothesis,∈ there exists an idempotent{ } e A such− that{ e/1} = f. It follows that e p and 1 e q. ∈ (ii∈) (vii)− : By∈ Theorem 2.3, the set of flat clopens of Min(A) forms a basis⇒ for the induced flat topology on Min(A). If U Min(A) is a flat clopen, then by Theorem 2.2, there exists an idempotent⊆ e A such that γ−1(U) = V (e) where γ : Spec(A) Min(A) is the retraction∈ map. It follows that U = V (e) Min(A). → (vii) (i) : If p and q are distinct∩ minimal primes of A, then U = Spec(A⇒) q is an open neighborhood of p, because the closed points of Spec(A\{) with} respect to the flat topology are precisely the minimal prime ideals of A. Thus by the hypothesis, there exists an idempotent e A such that p V (e) Min(A) U Min(A). It follows that e ∈ p and 1 e q.∈ ∩ ⊆ ∩ (vii∈ ) (viii−) :∈ If I is a pure ideal of A, then V (I) is a flat closed ⇒ subset of Spec(A). Thus by the hypothesis, there exists a set ei of idempotents of A such that U Min(A)= V (e ) Min(A) {where} ∩ i ∩ i U = Spec(A) V (I). It follows that U = SV (e ), because U is stable \ i Si under the specialization. Therefore V (I) = D(ei) = V (J) where Ti the ideal J is generated by the 1 ei. It follows that I = J, see [27, Theorem 3.2]. − (viii) (iii): If p is a minimal prime of A, then by Theorem 6.4, it is a pure⇒ ideal. Thus by the hypotheses, p is a regular ideal of A. (iv) (ix) : If M is a max-regular ideal of A, then V (M) is a connected⇒ component of Spec(A). Thus by the hypothesis, there exists a minimal prime p of A such that V (M) = V (p). It follows that M √M = p. But p is a regular ideal of A, see (iii). This yields that M ⊆= p. By a similar argument, it is shown that every minimal prime ideal of A is a max-regular ideal of A. (ix) (iv) : For any ring A, the connected components of Spec(A) are precisely⇒ of the form V (M) where M is a max-regular ideal of A. (viii) (x) : By Theorem 6.4, Ann(f) is a pure ideal. Thus by the hypothesis,⇒ it is a regular ideal. (x) (xi) : By Lemma 8.4, there exists an idempotent e Ann(g) such⇒ that f = fe. It follows that g = g(1 e). ∈ (xi) (i) : Let p and q be distinct minimal− prime ideals of A. By Lemma⇒ 3.2, there exist f A p and g A q such that fg = 0. Then ∈ \ ∈ \ GELFAND RINGS, CLEAN RINGS AND THEIR DUAL RINGS 27 by the hypothesis, there is an idempotent e A such that f = fe and g = g(1 e). This yields that 1 e p and∈e q. − − ∈ ∈ The following result provides an elementwise characterization for reduced purified rings.

Corollary 8.6. Let A be a ring. Then A is a reduced puriﬁed ring if and only if fg = 0 for some f,g A, then there exists an idempotent e A such that f = fe and g = g∈(1 e). ∈ −

Proof. For the implication “ ” see Theorem 8.5 (xi). Conversely, if f A is nilpotent then we may⇒ choose the least natural number n > 1 such∈ that f n =0. If n> 1 then there exists an idempotent e A such that f = fe and f n−1 = f n−1(1 e), but f n−1 = f n−2f(1 ∈ e)=0 which is a contradiction. So f =− 0. Hence, A is reduced.− Thus by Theorem 8.5 (xi), A is also a puriﬁed ring.

Corollary 8.7. The direct product of a family of rings is a reduced puriﬁed ring if and only if each factor is a reduced puriﬁed ring.

Proof. It follows from Corollary 8.6.

There are reduced purified rings which are not p.p. rings, (recall that a ring A is called a p.p. ring if each principal ideal of A is a projective A module, or equivalently, Ann(f) is generated by an idempotent el- − ement for all f A). For example, the ring C(βN N) is a reduced purified ring which∈ is not a p.p. ring. There are also\ reduced mp-rings which are not purified rings. For instance, the ring C(βR+ R+) is a reduced mp-ring which is not a purified ring. For the details\ see [3]. Note that by Theorem 8.5 (x), reduced purified rings and almost p.p. rings (in the sense of [3]) are the same. Here, by C(X) we mean the ring of real-valued continuous functions on X, and βX denotes the Stone-Cechˇ compactification of the topological space X. Hence, the following inclusions are strict: p.p. rings reduced purified rings p.f. rings = reduced mp-rings. ⊂ ⊂ Acknowledgements. The authors would like to give sincere thanks to Professors Pierre Deligne and Fran¸cois Couchot for very valuable cor- respondences during the writing of the present paper. We would also like to give heartfelt thanks to the referees for very careful reading of 28 M.AGHAJANIANDA.TARIZADEH the paper and for their very valuable comments and suggestions which improved the paper.

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Department of Mathematics, Faculty of Basic Sciences, University of Maragheh, P. O. Box 55136-553, Maragheh, Iran. Email address: [email protected], [email protected]