Rings in which the unique primary decomposition theorem holds

Abstract

Every ring considered in this paper will be assumed to be commutative and to contain an identity element. A ring R will be said to be a W-ring (or to have property W) if each ideal of R may be uniquely represented as an intersection of finitely many primary ideals. A ring R will be called a W*-ring (or to have property W*) if R is a W-ring and every ideal of R contains a power of its radical. A quasi-local ring is a ring with exactly one maximal ideal. An integral domain D is strongly integrally closed if it has the following property: If x is an element of the quotient field K of D such that all powers of x are in a finite D-module contained in K, then xED. This paper gives necessary and sufficient conditions that a ring be a W-ring. The ideal theory of W-rings and W*-rings is also investigated. Mori [3] has given necessary and sufficient conditions for a ring to be a W-ring, but his characterization of such rings is fairly involved due to the fact that he does not assume his ring has an identity. The terminology used in this paper is that of Zariski and Samuel [6]. The symbol C is used to denote containment while C denotes proper containment. Some elementary properties of W-rings and W*-rings are these: PROPERTY 1. If R is a W-ring (respectively, W*-ring) and A is an ideal of R, then RIA is a W-ring (W*-ring) [6, pp. 148-154]. PROPERTY 2. If ring R has property W (respectively, W*) and M is a multiplicative system of R such that 0 {M, then RM, the ring of quotients of R with respect to M, has property W (W*). In particular, if P is a proper prime ideal of R, then the quotient ring Rp of R with respect to P has property W (W*) [6, p. 225]. PROPERTY 3. A finite direct sum of rings is a W-ring (W*-ring) if and only if each summand is a W-ring (W*-ring) [6, p. 175].