Generalized primary rings

Abstract

The purpose of this paper is to investigate the properties of commutative rings with identity in which all ideals are primary. We call such rings generalized primary rings. Zariski primary rings [8; p. 204], Snapper's completely primary rings [7] and discrete valuation domains are generalized primary. It is shown in the example constructed in 2.4 that generalized primary rings need not be primary rings. There are also examples of generalized primary rings which are not discrete valuation rings (4.11). The principal results concerning generalized primary rings are summarized here. Generalized primary rings are local rings (Theorem 2.3) and a Noetherian domain is generalized primary iff it is a one dimensional local ring (Theorem 4.5). If the requirement that the ring is a domain is dropped, then this result need not be true (4.7). A commutative hereditary ring with identity is generalized primary iff it is a discrete valuation domain (Theorem 4.9). A Noetherian generalized primary is a discrete valuation ring iff the unique maximal ideal is principal (Theorem 4.11). The importance of the first two results mentioned above can be realized by noting that the ring of integers is one dimensional Noetherian hereditary domain but is not a generalized primary ring. Most of the results in this paper are obtained by using the concept of the radical. For this we refer the reader to [8; p. 147]. V ~ represents the radical of an ideal A and ] /~ is the intersection of all prime ideals containing A E2; p. 253].