On weakly quasi-$S$-primary ideals
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This article introduces the concept of weakly quasi-$S$-primary ideals, which generalize both weakly $S$-primary and quasi-$S$-primary \linebreak ideals. We define an ideal $P$ of a ring $R$ to be {\em weakly quasi-$S$-primary} if there exists $s \in S$ such that for all $x, y \in R$ if $0 \neq xy \in P$, then either $sx \in \sqrt{P}$ or $sy \in \sqrt{P}$. We investigate various properties and characterizations of weakly quasi-$S$-primary ideals, and provide numerous examples and counterexamples. We also study the behavior of these ideals under homomorphisms, in rings of fractions, in idealizations, and in amalgamated rings.
Abstract
This article introduces the concept of weakly quasi-$S$-primary ideals, which generalize both weakly $S$-primary and quasi-$S$-primary \linebreak ideals. We define an ideal $P$ of a ring $R$ to be {\em weakly quasi-$S$-primary} if there exists $s \in S$ such that for all $x, y \in R$ if $0 \neq xy \in P$, then either $sx \in \sqrt{P}$ or $sy \in \sqrt{P}$. We investigate various properties and characterizations of weakly quasi-$S$-primary ideals, and provide numerous examples and counterexamples. We also study the behavior of these ideals under homomorphisms, in rings of fractions, in idealizations, and in amalgamated rings.