On wsq-primary ideals

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 ≠ ab ∈ Q for some a, b ∈ R, then a2 ∈ Q or b∈Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b \in \sqrt Q $$\end{document}. 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.