Related papers: A note on lower nil M-Armendariz rings

We introduce a complete radical formula for modules over non-commutative rings which is the equivalence of a radical formula in the setting of modules defined over commutative rings. This gives a general frame work through which known…

Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of weakly $1$-absorbing prime ideals which is a generalization of weakly prime ideals. A proper ideal $I$ of $R$ is called weakly $1$-absorbing prime if…

A monomial ideal $I\subseteq \mathbb{K}[x_1,\ldots , x_n]$ is called a Simis ideal if $I^{(s)}=I^s$ for all $s\geq 1$, where $I^{(s)}$ denotes the $s$-th symbolic power of $I$. Let $I$ be a support-2 monomial ideal such that its irreducible…

Let $B$ be a reduced local (Noetherian) ring with maximal ideal $M$. Suppose that $B$ contains the rationals, $B/M$ is uncountable and $|B| = |B/M|$. Let the minimal prime ideals of $B$ be partitioned into $m \geq 1$ subcollections $C_1,…

Let $R$ be a commutative ring with a non-zero identity, $S$ be a multiplicatively closed subset of $R$ and $M$ be a unital $R$-module. In this paper, we define a submodule $N$ of $M$ with $(N:_{R}M)\cap S=\phi$ to be weakly $S$-prime if…

It is proved that if $D$ is a $UFD$ and $R$ is a $D$-algebra, such that $U(R)\cap D\neq U(D)$, then $R$ has a maximal subring. In particular, if $R$ is a ring which either contains a unit $x$ which is not algebraic over the prime subring of…

In this paper, we introduce the concepts of 1-absorbing prime and weakly 1-absorbing prime subsemimodules over commutative semirings. Let S be a commutative semiring with 1 \neq 0 and M an S-semimodule. A proper subsemimodule N of M is…

Let $R$ be a commutative ring with $ 1 \neq 0$. We recall that a proper ideal $I$ of $R$ is called a semiprimary ideal of $R$ if whenever $a,b\in R$ and $ab \in I$, then $a\in \sqrt{I}$ or $b\in \sqrt{I}$. We say $I$ is a {\it weakly…

Let $R$ be a commutative ring. When is a subgroup of $(R, +)$ an ideal of $R$? We investigate this problem for the rings $\mathbb{Z}^{d}$ and $\prod_{i=1}^{d} \mathbb{Z}_{n_{i}}$. For various subgroups of these rings we obtain necessary and…

In this note we find the least order of a noncommutative even square ring and we note that it is a nil ring having characteristic four. In order to prove the main result given in this note we mainly use suitable examples.

Throughout this paper, $R$ is an associative ring (not necessarily commutative) with identity and $M$ is a right $R$-module with unitary. In this paper, we introduce a new concept of $\phi$-prime submodule over an associative ring with…

Let $R$ be a commutative ring with identity. The ring $R\times R$ can be viewed as an extension of $R$ via the diagonal map $\Delta: R \hookrightarrow R\times R$, given by $\Delta(r) = (r, r)$ for all $r\in R$. It is shown that, for any $a,…

This is the second in a series of papers highlighting the applications of reduced and coreduced modules. Let $R$ be a commutative unital ring and $I$ be an ideal of $R$. We give necessary and sufficient conditions in terms of $I$-reduced…

We will explore some properties of minimal graded free resolutions of $R/I$, where $R$ is a trivariate polynomial ring over a field and $I$ is a monomial ideal. Our focus will be to consider a specific form of the resolutions when $I$ is…

We study the algebraic and arithmetic structure of monoids of invertible ideals (more precisely, of $r$-invertible $r$-ideals for certain ideal systems $r$) of Krull and weakly Krull Mori domains. We also investigate monoids of all nonzero…

We characterize when the monomial maximal ideal of a simplicial affine semigroup ring has a monomial minimal reduction. When this is the case, we study the Cohen-Macaulay and Gorenstein properties of the associated graded ring and provide…

For a monomial ideal $I$, let $G(I)$ be its minimal set of monomial generators. If there is a total order on $G(I)$ such that the corresponding Lyubeznik resolution of $I$ is a minimal free resolution of $I$, then $I$ is called a Lyubeznik…

Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing "mesoprimary decompositions" determined by their underlying monoid congruences. These mesoprimary…

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…

We study certain (two-sided) nil ideals and nilpotent ideals in a Lie nilpotent ring R. Our results lead us to showing that the prime radical rad(R) of R comprises the nilpotent elements of R, and that if L is a left ideal of R, then…