On Weakly δ-Semiprimary Ideals of Commutative Rings

Abstract

Let R be a commutative ring with 1 not equal 0. A proper ideal I of R is a semiprimary ideal of R if whenever a, b is an element of R and ab is an element of I, we have a & nbsp;is an element of root I or b is an element of root I; and I is a weakly semiprimary ideal of R if whenever a, b is an element of R and 0 not equal ab is an element of I, we have a is an element of root I or b is an element of root I. In this paper, we introduce a new class of ideals that is closely related to the class of (weakly) semiprimary ideals. Let I(R) be the set of all ideals of R and let delta : I(R) -> I(R) be a function. Then delta is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J subset of I, we have L subset of S(L) and S(J) subset of delta(I). Let delta be an expansion function of ideals of R. Then a proper ideal I of R is called a delta-semiprimary (weakly delta-semiprimary) ideal of R if ab is an element of I (0 not equal ab is an element of I) implies a is an element of S(I) or b is an element of delta(I). A number of results concerning weakly delta-semiprimary ideals and examples of weakly delta-semiprimary ideals are given.