Rings and radicals related to n-primariness

Abstract

This paper concerns ring properties which are induced from the structure of the powers of prime ideals. An ideal I of a ring R is called n-primary (respectively, T-primary) provided that AB ⊆ I for ideals A, B of R implies that (A + I)/I or (B + I)/I is nil of index n (respectively, (A + I)/I or (B + I)/I is nil) in R/I, where n ≥ 1. It is proved that for a proper ideal I of a principal ideal domain R, I is T-primary if and only if I is of the form pkR for some prime element p and k ≥ 1 if and only if I is 2-primary, through which we study the structure of matrices over principal ideal domains. We prove that for a T-primary ideal I of a ring R, R/I is prime when the Wedderburn radical of R/I is zero. In addition we provide a method of constructing strictly descending chain of n-primary radicals from any domain, where the n-primary radical of a ring R means the intersection of all the n-primary ideals of R. © World Scientific Publishing Company