Right primary and nilary rings and ideals

Recall that in a commutative ring R an ideal I is called primary if whenever a,b∈R with ab∈I then either a∈I or bn∈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.