Right primary and nilary rings and ideals

Recall that in a commutative ring R an ideal I is called primary if whenever a, b is an element of R with ab is an element of 1 then either a is an element of I or b(n) is an element of I, for some positive integer n. A commutative ring R is called primary if the zero ideal is a primary ideal. In this paper, we investigate various generalizations of the primary concept to noncommutative rings. In particular, we determine conditions on a ring R such that: (1) each ideal of R is a finite intersection of ideals satisfying one of the generalizations of the primary concept; or (2) R is a finite direct. sum of rings satisfying one of the generalizations of the primary concept; or (3) R has a generalized triangular matrix representation in which each ring on the main diagonal satisfies one of the generalizations of the primary concept. Examples are provided to illustrate and delimit our results. (C) 2013 Elsevier Inc. All rights reserved.