Related papers: Weakly J-ideals of Commutative Rings

Can there be a structure space-type theory for an arbitrary class of ideals of a ring? The ideal spaces introduced in this paper allows such a study and our theory includes (but not restricted to) prime, maximal, minimal prime, strongly…

It is proven that each commutative arithmetical ring $R$ has a finitistic weak dimension $\leq 2$. More precisely, this dimension is 0 if $R$ is locally IF, 1 if $R$ is locally semicoherent and not IF, and 2 in the other cases.

Let $G$ be a group with identity $e$ and $R$ be a $G$-graded commutative ring with nonzero unity $1$. In this article, we introduce the concept of graded strongly $1$-absorbing primary ideals. A proper graded ideal $P$ of $R$ is said to be…

Let J \subseteq I be ideals in a commutative Noetherian ring R, and r,s \geq 0. We say that J is a demotion of I if I^r J^s = I^{r+s} \cap J^s for all r,s \geq 0. In this paper, we mainly aim to explore this notion in polynomial rings. In…

In this paper, the $\tau_q$-weak global dimension $\tau_q$-\cwd$(R)$ of a commutative ring $R$ is introduced. Rings with $\tau_q$-weak global dimension equal to $0$ are studied in terms of homologies, direct products, polynomial extensions…

The annihilating-ideal graph of a commutative ring $R$ with unity is defined as the graph $\mathbb{AG}(R)$ with the vertex set is the set of all non-zero ideals with non-zero annihilators and two distinct vertices $I$ and $J$ are adjacent…

Given a commutative ring $R$ and finitely generated ideal $I$, one can consider the classes of $I$-adically complete, $L_0^I$-complete and derived $I$-complete complexes. Under a mild assumption on the ideal $I$ called weak pro-regularity,…

In this paper, we introduce a class of rings which is a generalization of reflexive rings and $J$-reversible rings. Let $R$ be a ring with identity and $J(R)$ denote the Jacobson radical of $R$. A ring $R$ is called {\it $J$-reflexive} if…

In this paper, we introduce $\phi$-1-absorbing prime ideals in commutative rings. Let $R$ be a commutative ring with a nonzero identity $1\neq0$ and $\phi:\mathcal{I}(R)\rightarrow\mathcal{I}(R)\cup\{\emptyset\}$ be a function where…

As pointed out in the monographs by J. S. Golan and by W. Kuich and A. Salomaa on semirings, ideals play an important role despite the fact that they need not be congruence kernels as in the case of rings. Hence, having two commutative…

Let R be a commutative ring with identity and M be an R-module. A proper ideal I of R is said to be a $z^\circ$-ideal if for each $a \in I$ the intersection of all minimal prime ideals containing a is contained in I. The purpose of this…

Let R be a multiplicative hyperring with identity. In this paper, we define the concept of J-prime hyperideals which is a generalization of n-hyperideals and we will show some properties of them. Then we extend the notion of J-prime to…

In this study, all rings are commutative with non-zero identity and all modules are considered to be unital. Let $M$ be a left $R$-module. A proper submodule $N$ of $M$ is called an $S$-$weakly$ $prime$ submodule if $0_{M}\neq f(m)\in N$…

Let R be an associative ring with possible extra structure. R is said to be weakly small if there are countably many 1-types over any finite subset of R. It is locally P if the algebraic closure of any finite subset of R has property P. It…

For $a\in R$, let $P_a$ denote the intersection of all minimal prime ideals of $R$ containing $a$. An ideal $I$ of a ring $R$ is called a $z^{\circ}$-ideal if $P_a\subseteq I$ for all $a\in I$. In this paper, we first investigate the class…

Let $R$ a commutative ring, $\mathfrak{a} \subset R$ an ideal, $I$ an injective $R$-module and $S \subset R$ a multiplicatively closed set. When $R$ is Noetherian it is well-known that the $\mathfrak{a}$-torsion sub-module…

Researchers introduced the notion of j-Artinian rings in [3] and obtained significant results concerning this new class of rings. Motivated by their definition and findings, we extend the study to modules by introducing the concept of…

We introduce and develop the theory of weakly Arf rings, which is a generalization of Arf rings, initially defined by J. Lipman in 1971. We provide characterizations of weakly Arf rings and study the relation between these rings, the Arf…

This paper considers graded near-rings over a monoid G as a generalizations of the graded rings over groups, introduce certain innovative graded weakly prime ideals and graded almost prime ideals as a generalizations of graded prime ideals…

Let $M$ be G-graded R-module. The idea of a graded weakly primal submodule of $M$, which is a generalization of a graded primal submodule, is introduced and discussed in this paper. Some characteristics and characterizations are assigned to…