Journal of Pure and Applied Algebra 146 (2000) 283–290 www.elsevier.com/locate/jpaa View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Characterizations of Krull rings with zero divisors1 Byung Gyun Kang ∗ Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, South Korea Communicated by C.A. Weibel; received 12 February 1998; received in revised form 5 May 1998 Abstract We show that a ring is a Krull ring if and only if every nonzero regular prime ideal contains a t-invertible prime ideal if and only if every proper regular principal ideal is quasi-equal to a product of prime ideals. c 2000 Elsevier Science B.V. All rights reserved. MSC: 13A05; 13A15; 13E99 1. Introduction It is a well-known fact that an integral domain is a unique factorization domain if and only if every nonzero prime ideal contains a prime. Analogously, an integral domain is a Krull domain if and only if every nonzero prime ideal contains a t-invertible prime ideal [6]. For rings with zero divisors, the author generalized the previous result [8]: a Marot ring is a Krull ring if and only if every regular prime ideal contains a t-invertible prime ideal. One of the purposes of this paper is to eliminate the ‘Marot’ hypothesis in [8, Theorem] and establish that a ring is a Krull ring if and only if every regular prime ideal contains a t-invertible prime ideal. In [12] Tramel showed that if every nonzero proper principal ideal of an integral domain R is quasi-equal to a product of prime ideals, then R is a Krull domain. Another purpose is to extend this result to rings with zero divisors. As in [8], we adopt Kennedy’s deÿnition of a Krull ring [9]. Thus we do not assume a Krull ring to be a Marot ring. There is another deÿnition of a Krull ring [3, 11], which is precisely a Marot Krull ring. In this paper R shall denote ∗ E-mail address: [email protected] (B.G. Kang) 1 This research was supported by the BSRI Research Grant 98-1431. 0022-4049/00/$ - see front matter c 2000 Elsevier Science B.V. All rights reserved. PII: S0022-4049(98)00100-5 284 B.G. Kang / Journal of Pure and Applied Algebra 146 (2000) 283{290 a commutative ring with identity and with total quotient ring T(R):Z(R) stands for the set of zero divisors of R and an element r 2 reg(R):=RnZ(R) is called a regular element of R. An ideal of R is called a regular ideal if it contains a regular element. R is called a Krull ring if there exists a family f(V ;P )j 2g of discrete rank one valuation pairs of T(R) with associated valuations fv j 2 g such that T (1) R = fV j 2 g, and (2) v (ÿ) = 0 almost everywhere on for each regular element ÿ 2 T(R), and each P is a regular ideal of V . For details, see [3]. The R-submodules of T(R) will be denoted by F(R). For I 2 F(R);I−1 is deÿned to be [R : I] ≡{j2T(R);I⊆Rg. The v-operation on F(R) P is deÿned as usual: Iv =[R:[R:I]]. For I 2 F(R);It is deÿned to be (I0)v, where I0 runs over the ÿnitely generated R-submodules contained in I. We say that I is −1 t-invertible if (II )t = R. Conventionally, the inverse, the v-operation, and the t- operation are deÿned for regular fractional ideals [2]. However, these for R-submodules of T(R) share many useful properties with those for regular fractional ideals of R [4, Lemma 3.3]. 2. Preliminaries We emphasize that F(R) is the set of R-submodules of T(R). A ∗-operation on F(R) is a mapping I 7! I∗ of F(R) into F(R) such that for each a 2 T(R) and for each I; J 2 F(R), (I) aI∗ ⊆ (aI)∗, (II) I ⊆ I∗; I ⊆ J implies I∗ ⊆ J∗, (III) (I∗)∗ = I∗. From (I)–(III), the following directly follow for fI j 2 }⊆F(R). P P (IV) ( I )∗ =( (I )∗)∗, (V) (IJ) =(IJ ) , T ∗ ∗ ∗T (VI) ( (I )∗)∗ = (I )∗. It is routine to see that the v-operation and t-operation are ∗-operations. We call I 2 F(R)a∗-module if I∗ = I. In case when I is an ideal of R and I∗ = I, we say that I is a ∗-ideal of R. As in the domain case, any prime ideal minimal over a ∗-ideal is a ∗-ideal provided that ∗ is of ÿnite type. Recall that ∗ is of ÿnite type if for every P ideal I of R; I∗ = f(I0)∗ j I0 is a ÿnitely generated ideal of R contained in Ig. For a ∗-operation of ÿnite type, it is easy to show that any proper ∗-ideal is contained in a maximal ∗-ideal and a maximal ∗-ideal is a prime ideal. A typical example of a ÿnite type ∗-operation is the t-operation. It may happen that R = T(R) and in this case R is both the unique v-ideal and the unique t-ideal of R. To avoid this peculiar case, we assume that R 6= T(R), i.e., R contains a nonunit regular element. Each nonunit regular element provides a proper v-ideal and a maximal t-ideal containing it. For f 2 R[X ], let Af be the ideal of R generated by the coecients of f. B.G. Kang / Journal of Pure and Applied Algebra 146 (2000) 283{290 285 In the domain case, the multiplicative subset Nv = ff j f 2 R[X ] and (Af)v = Rg is proven to be useful in several papers [5–7]. There are two possible generalizations of Nv to the ring of zero divisors. One is ff 2 R[X ] j (Af)v = Rg, which will be denoted by Nv(R). The other is ff 2 R[X ] j Af is a regular ideal of R and (Af)v = Rg, which r will be denoted by N (R). We write RhX i := R[X ] r . Let maxt(R) and rmaxt(R) v Nv (R) denote the set of maximal t-ideals and regular maximal t-ideals of R respectively. We denote by P2 the set fP 2 Spec(R) j P is minimal over (a : b);a;b2R with a regularg. ;Nr; max(R[X ] ), and max(RhX i). We give a description of the sets Nv v Nv Proposition 1. (1) N and N r are saturated multiplicative subsets of R[X ]. S v v S S (2) Nv = R[X ]n M [X ]=R[X]n M [X ]=R[X]n P[X]. M∈maxt(R) M∈rmaxt(R) P∈P2 r r (3) Nv ⊆ Nv (the inclusion can be proper), Nv ⊆ reg(R[X ]); and Nv might contain a zero divisor. (4) max(R[X ] )=fM[X] jM2maxt(R)g; max(RhX i)⊇{M[X] r jM 2rmaxt(R)g; Nv Nv Nv and the inclusion can be proper. n+1 n Proof. (1) Let f; g 2 Nv. For some n ≥ 1;Af Ag = AfAfg by the Dedekind–Mertens n+1 lemma : Corollary (28.3) in [2]. Applying v-operations, we get R =((Af)v (Ag)v)v = n+1 n n (Af Ag)v =(AfAfg)v =((Af)v(Afg)v)v =(Afg)v. Thus fg 2 Nv. If in addition Af;Ag are n+1 regular ideals of R, let a 2 Af \ reg(R) and b 2 Ag \ reg(R). Note that a b 2 Afg. Thus fg 2 N r. v S (2) For the equality N = R[X ]n M [X ], imitate the domain case proof. S v M∈maxtS (R) S Clearly, R[X ]n M∈maxt(R) M [X ] ⊆ R[X ]n M∈rmaxt(r) M [X ]. Let f 2 R[X ]n M∈rmaxt(R) M [X ] and suppose that (Af)v 6= R. Choose a=b 2 T(R) with b 2 reg(R) such that (a=b) Af ⊆ R.SoAf⊆[b:a] and [b : a] is a regular t-ideal of R. Enlarging this to a regular maximal t-ideal M of R,wegetA ⊆M, i.e., f 2 M [X ], a desired contradiction. Thus S f R[X ]n M∈rmaxt(R) M [X ] ⊆ Nv. For the third equality, see [3, Theorem 19.1]. r (3) Let R = Z ⊕ Q and f =(1;0)+(1;0)X 2 R[X ]. Then f 2 NvnNv : Af =(1;0); M = Z ⊕ 0 is a maximal ideal of R; Af ⊆ Z(R), and (Af)v = R [3, p121]. Moreover r f 2 Z(R[X ]). Let g 2 Nv .Ifg2Z(R[X]), then 9b 2 Rnf0g such that bg = 0. But then bAg = 0 and hence b = 0, a contradiction. So g 2 reg(R[X ]). (4) For the equality, imitate the domain case. If max(RhX i)=fM[X]Nr jM2rmaxt S v r r (R)g, then Nv = R[X ]n M∈rmaxt(R) M [X ]=Nv since Nv is a saturated multiplicative subset of R[X ], which is not the case (see (2) and (3)). I 2 F(R) is said to be invertible if IJ = R for some J 2 F(R) or equivalently II −1 = R. It is easy to see that an invertible ideal is ÿnitely generated and locally principal.