δ-primary Ideals of Commutative Rings

In this paper we investigate -primary ideals which unify prime ideals and primary ideals. A number of main results about prime ideals and primary ideals are extended into this general framework. Prime ideals and primary ideals are two of the most important structures in commutative algebra. Although different from each other in many aspects, they share quite a number of similar properties as well( see [1] ). However, these two structures have been treated rather differently, and all of their properties were proved separately. It is therefore natural to examine whether it is possible to have a unified approach to studying these two structures. In this short paper we introduce the notion of -primary ideals where is a mapping that assigns to each ideal I an ideal (I) of the same ring. Such -primary ideals unify the prime and primary ideals under one frame. This approach clearly reveals how similar the two structures are and how they are related to each other. In the first section, we introduce ideal expansion and define primary ideals with respect to such an expansion. Besides the familiar expansions 0, 1 and B, we also have a new expansion M defined by means of maximal ideals. In the second section, we investigate ideal expansions satisfying some additional conditions and prove more properties of the generalized primary ideals with respect to such expansions. In this paper, all the rings used are commutative rings with an multiplication identity and all the ring homomorphisms preserve the identity. We shall use Id(R) to denote the set of all ideals of the ring R.