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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 $0\ne ab\in Q$ for some $a,b\in R$, then $a^2\in Q$ or $b\in \sqrt{Q}.$ 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...

Recently, motivated by Anderson, Dumitrescu’s $S$-finiteness, D. Bennis, M. El Hajoui (2018) introduced the notion of $S$-coherent rings, which is the $S$-version of coherent rings. Let $R={⨁}_{\alpha \in G}{R}_{\alpha }$ be a commutative ring with unity graded by an arbitrary commutative monoid $G$, and $S$ a multiplicatively closed subset of nonzero homogeneous elements of $R$. We define $R$ to be graded-$S$-coherent ring if every finitely generated homogeneous ideal of $R$ is $S$-finitely presented. The purpose of this paper is to give the graded...