Narrow Results

We explore the concept of the $S$-Krull dimension in commutative rings, extending classical notions of Krull dimension by incorporating multiplicative subsets $S$ of a ring $R$. We provide characterizations of $S$-prime, $S$-maximal, and $S$-minimal $S$-prime ideals, establishing foundational results crucial for understanding the $S$-Krull dimension. The paper also delves into properties of $S$-principal ideal $S$-domains, $S$-Artinian rings, and u-$S$-von Neumann regular rings, introducing an $S$-variant of the Division Algorithm. In the final section, the concept of $S$-height for $S$-prime ideals is defined, and the $S$-Krull dimension is analyzed through various characterizations and examples, with a particular focus on polynomial rings. We conclude with insights into the $S$-height of larger $S$-prime ideals of $R[X]$ and their intersections with $R$, contributing to a deeper understanding of the $S$-Krull dimension.

Let $R$ be a commutative ring with identity and $S$ be a multiplicatively closed subset of $R$. The purpose of this paper is to introduce the concept of weakly $S$-primary ideals as a new generalization of weakly primary ideals. An ideal $I$ of $R$ disjoint with $S$ is called a weakly $S$-primary ideal if there exists $s\in S$ such that whenever $0\ne ab\in I$ for $a,b\in R$, then $sa\in \sqrt{I}$ or $sb\in I$. The relationships among $S$-prime, $S$-primary, weakly $S$-primary and $S$-$n$-ideals are investigated. For an element $r$ in any general ZPI-ring, the (weakly) $S_r$-primary ideals are characterized where $S_r=\{1,r,r^2,\dots \}$. Several properties, characterizations and examples concerning weakly $S$-primary ideals are presented. The stability of this new concept with respect to various ring-theoretic constructions such as the trivial ring extension and the amalgamation of rings along an ideal are studied. Furthermore, weakly $S$-decomposable ideals and $S$-weakly Laskerian rings which are generalizations of $S$-decomposable ideals and $S$-Laskerian rings are introduced.

Let $R$ be a commutative super-ring with unity $(1\ne 0)$ and let $𝔍(R)$ be the set of all superideals of $R$. Let $\varphi :𝔍(R)\to 𝔍(R)\cup \{\varnothing \}$ be a reduction function of superideals of $R$ and let $\delta :𝔍(R)\to 𝔍(R)$ be an expansion function of superideals of $R$. We recall that a proper superideal $I$ of $R$ is called a $\varphi$-1-absorbing $\delta$-primary superideal of $R$, if whenever $abc\in I\backslash \varphi (I)$ for some nonunit elements $a,b,c\in h(R)$, then $ab\in I$ or $c\in \delta (I)$. In this paper, we introduce a new class of superideals that is a generalization to the class of $\varphi$-1-absorbing $\delta$-primary superideals. Let $S$ be a multiplicative subset of the subset $h(R)$ of the homogeneous elements such that $1\in S$ and let $I$ be a proper superideal of $R$ with $S\cap I=\varnothing$, then $I$ is called a $\varphi$-$S$-1-absorbing $\delta$-primary superideal of $R$ associated to $s\in S,$ if whenever $abc\in I\backslash \varphi (I)$ for some nonunit elements $a,b,c\in h(R)$, then $sab\in I$ or $sc\in \delta (I)$. In this paper, we have presented a range of different examples, properties and characterizations of this new class of superideals.