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In this paper, we study multiplication lattice modules. We establish a new multiplication over elements of a multiplication lattice module.With this multiplication, we characterize idempotent element, prime element, weakly prime element and almost prime element in multiplication lattice modules.

Let $R$ be a commutative ring with a nonzero identity. In this study, we present a new class of ideals lying properly between the class of $n$-ideals and the class of $(2,n)$-ideals. A proper ideal $I$ of $R$ is said to be a quasi $n$-ideal if $\sqrt{I}$ is an $n$-ideal of $R.$ Many examples and results are given to disclose the relations between this new concept and others that already exist, namely, the $n$-ideals, the quasi primary ideals, the $(2,n)$-ideals and the $pr$-ideals. Moreover, we use the quasi $n$-ideals to characterize some...

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...