On wsq-primary ideals

Abstract

We introduce weakly strongly quasi-primary (briefly, wsq-primary) ideals in commutative rings. Let R be a commutative ring with a nonzero identity and Q a proper ideal of R. The proper ideal Q is said to be a weakly strongly quasi-primary ideal if whenever 0 &NOTEQUexpressionL; ab is an element of Q for some a, b is an element of R, then a(2) is an element of Q or b is an element of root Q. Many examples and properties of wsq-primary ideals are given. Also, we characterize nonlocal Noetherian von Neumann regular rings, fields, nonlocal rings over which every proper ideal is wsq-primary, and zero dimensional rings over which every proper ideal is wsq-primary. Finally, we study finite union of wsq-primary ideals.