COLLOQUIUMMATHEMATICUM VOL. 83 2000 NO. 1 ON THE MAXIMAL SPECTRUM OF COMMUTATIVE SEMIPRIMITIVE RINGS BY K. S A M E I (HAMADAN) Abstract. The space of maximal ideals is studied on semiprimitive rings and reduced rings, and the relation between topological properties of Max(R) and algebric properties of the ring R are investigated. The socle of semiprimitive rings is characterized homologically, and it is shown that the socle is a direct sum of its localizations with respect to isolated maximal ideals. We observe that the Goldie dimension of a semiprimitive ring R is equal to the Suslin number of Max(R). 1. Introduction. Throughout this paper, R is a commutative ring with identity. We write Spec(R), Max(R) and Min(R) for the spaces of prime ide- als, maximal ideals and minimal prime ideals of R, respectively. The topol- ogy of these spaces is the Zariski topology (see [2], [4], [5] and [7]). Also we denote by P0(R), M0(R) and I0(R) the sets of isolated points of the spaces Spec(R), Max(R) and Min(R), respectively. We say R is semiprimitive if T Max(R) = (0). For a semiprimitive Gelfand ring R, we show that P0(R) = M0(R) = I0(R) = Ass(R): A nonzero ideal in a commutative ring is said to be essential if it inter- sects every nonzero ideal nontrivially, and the intersection of all essential ideals, or the sum of all minimal ideals, is called the socle (see [9]). We characterize the socle of semiprimitive rings in two ways: in terms of maxi- mal ideals and in terms of localizations with respect to maximal ideals. We denote the socle of R by S(R) or S and the Jacobson radical of R by J(R). We know that the infinite intersection of essential ideals in any ring may not be an essential ideal. We shall show that in a semiprimitive ring, every intersection of essential ideals is an essential ideal if and only if M0(R) is dense in Max(R). A set fBigi2I of nonzero ideals in R is said to be independent if Bi \ P P L ( i6=j2I Bj) = (0), i.e., i2I Bi = i2I Bi. We say R has a finite Goldie dimension if every independent set of nonzero ideals is finite, and if R does not have a finite Goldie dimension, then the Goldie dimension of R, denoted 2000 Mathematics Subject Classification: 13A18, 13B24, 13C10, 13C11, 54C40. [5] 6 K. SAMEI by dim R, is the smallest cardinal number c such that every independent set of nonzero ideals in R has cardinality less than or equal to c. Also the smallest cardinal c such that every family of pairwise disjoint nonempty open subsets of a topological space X has cardinality less than or equal to c is called the Suslin number or cellularity of the space X and is denoted by S(X) (see [3]). We show that for any semiprimitive ring R, the Suslin number of Max(R) is equal to the Goldie dimension of R. 2. The socle of semiprimitive rings Definition. Let M(a) = fM 2 Max(R): a 2 Mg for all a 2 R, and M(I) = fM(a): a 2 Ig for all ideals I of R. An ideal M 2 Max(R) is called trivial if M is generated by an idempotent element, i.e., M = (e), where e2 = e. Lemma 2.1. Suppose Nil(R) = J(R) and M is a maximal ideal of R. Then M = p(e), where e is an idempotent element if and only if M is an isolated point of Max(R). Furthermore in this case, if M = (e) and e 6= 0, then I = (1 − e) is a nonzero minimal ideal. P r o o f. Let M = p(e), where e2 = e. Then fMg = Max(R)−M(1−e), i.e., M is an isolated point of Max(R). Conversely, suppose fMg is open in Max(R). If Max(R) = fMg, then M = p(0). Otherwise, there exist T 0 a; b; r 2 R such that a 2 M 02Max(R)−{Mg M − M, b 2 M and ar + b = 1. Obviously, ab 2 J, hence (ab)n = 0 for some n > 0. We have 1 = (ar+b)2n = n n n a x1 + b x2. Let e = b x2. Then e(1 − e) = 0 and this means that e is an idempotent element of R. Also for every m 2 M, there is n > 0 such that [(1 − e)m]n = 0, so mn 2 (e), i.e., M = p(e). The following proposition is proved in [8, 1.6]. Lemma 2.2. If R is a semiprimitive ring then I is a nonzero minimal ideal of R if and only if I is contained in every maximal ideal except one, i.e., jM(I)j = 2. In [7], it is proved that the socle of C(X) (the ring of continuous func- tions) consists of all functions that vanish everywhere except at a finite number of points of X. We give a generalization of this fact. Theorem 2.3. In a semiprimitive ring R, the socle S = S(R) is exactly the set of all elements which belong to every maximal ideal except for a finite number. In fact, S = fa 2 R : Max(R) − M(a) is finiteg: P r o o f. Suppose a 2 S. If a = 0, then Max(R) − M(a) = ;. Otherwise, a = a1 + ::: + an, where each ai belongs to some idempotent minimal ideal MAXIMAL SPECTRUM 7 in R. Thus by 2.2, a1 + ::: + an belongs to every maximal ideal except for a finite number. It follows that Max(R) − M(a) is finite. Conversely, let Max(R) − M(a) be a finite set. We have to show that a 2 S. Let Max(R) − M(a) = fM1;:::;Mng. We claim that each Mk, k = 1; : : : ; n; is an isolated point of Max(R). Indeed, for every i 6= k and 1 ≤ i ≤ n there exists ai 2 Mi − Mk. Set b = aa1 : : : ak−1ak+1 : : : an. Then M(b) = Max(R) − fMkg, so fMkg is open, i.e., Mk is isolated. Now by 2.1, for each Mk, there exists a minimal ideal Ik such that R = Mk ⊕ Ik and Ik = (ek), where ek is an idempotent element of R. Let b = a − ae1 − ::: − aen. Then for every 1 ≤ k ≤ n, ekb 2 J(R) = 0, and consequently fM1;:::;Mng ⊆ M(b). On the other hand, M(a) ⊆ M(b), hence b = 0, therefore a = ae1 + ::: + aen 2 I1 + ::: + In ⊆ S. Now we give a characterization of the socle of semiprimitive rings by localizations with respect to maximal ideals. Theorem 2.4. Let R be a semiprimitive ring and let I be an ideal of R. Then I ⊆ S if and only if the sequence φ M π M 0 ! I ! IM ! IM ! 0 M2Max(R) M2Max(R)−M0(R) is exact, where φ is the natural map and π is the projection map and IM is the localization of I. Furthermore, the socle is the unique ideal with this property. P r o o f. ()) Suppose I ⊆ S, and consider the natural map φ : I ! L M2Max(R) IM such that 8a 2 I, φ(a) = (a=1)M2Max(R): Now suppose a 2 I. Then by 2.3, Max(R) − M(a) = fM1;:::;Mng. Hence for each 1 ≤ k ≤ n, there exists ek such that Mk = (ek). Put b = e1 : : : en. It is evident that ab 2 J = (0) and b 2 R − M for each M 2 M(a), so a=1 = 0 in IM . Hence φ is a well defined homomorphism. Also Ker φ = fa 2 I : 8M; 9t 2 R − M such that ta = 0g ⊆ J = (0); thus φ is one-to-one. Now we show that Im φ=Ker π. Suppose (b=t)M2Max(R) 2 Im φ. Then there is a 2 I such that (a=1)M2Max(R) = (b=t)M2Max(R). Ob- viously, a=1 = 0 in IM for every M 2 Max(R) − M0(R). (Since Max(R) − M(a) = fM1;:::;Mng, for each 1 ≤ k ≤ n there is tk 2 Mk − M. Let t = t1; : : : tn. Then t 2 R − M and at 2 J = (0).) Thus φ(a) 2 Ker π and consequently, Im φ ⊆ Ker π. To prove Im φ ⊇ Ker π, it is enough to show that if 0 6= b=t 2 IM , where M 2 M0(R), then there exists a 2 I such that the M-component of φ(a) is b=t and all the other components are zero. To see this, we note that b; t 62 M and M = (e) where e is an idempotent element of R. So there exists t0 2 R such that tt0 −1 2 M. Let a = (1−e)t0b and s = 1 − e. We have s(at − b) = sb(tt0 − 1 − tt0e) 2 J = (0). So a=1 = b=t 8 K. SAMEI in M-components, and also ea = 0, hence a=1 = 0 for all other components. Thus the sequence is exact. (() Let a 2 I and suppose the sequence φ M π M 0 ! I ! IM ! IM ! 0 M2Max(R) M2Max(R)−M0(R) is exact. Since φ(a) is well defined, every component of φ(a) is zero except for a finite numbers of components M1;:::;Mn. Clearly, Max(R)−M(a) ⊆ fM1;:::;Mng. Thus a 2 S, i.e., I ⊆ S.