Related papers: Square-difference factor absorbing ideals of a com…

We study the two dual notions of prime avoidance and prime absorbance. We generalize the classical prime avoidance lemma to radical ideals. A number of new criteria are provided for an abstract ring to be C.P. (every set of primes satisfies…

This paper mainly focuses on commutative local domains of dimension one. We then obtain a criterion for a ring to have a finite number of trace ideals in terms of integrally closed ideals. We also explore properties of such rings related to…

Let R be a commutative ring and I an ideal of R. A sub-ideal J of I is a reduction of I if JI^n = I^n+1 for some positive integer n. The ring R has the (finite) basic ideal property if (finitely generated) ideals of R do not have proper…

In this work, we introduce the notion of $S$-1-absorbing primary submodule as an extension of 1-absorbing primary submodule. Let $S$ be a multiplicatively closed subset of a ring $R$ and $M$ be an $R$-module. A submodule $N$ of $M$ with…

We call a ring R pointwise semicommutative if for any element a in R either l(a) or r(a) is an ideal of R. A class of pointwise semicommutative rings is a strict generalization of semicommutative rings. Since reduced rings are pointwise…

Perfect Gabriel filters of right ideals and their corresponding right rings of quotients have the desirable feature that every module of quotients is determined solely by the right ring of quotients. On the other hand, symmetric rings of…

Let S be a set of n ideals of a commutative ring A. Let G_{even} (respectively G_{odd}) denote the product of all the sums of even (respectively odd) number of ideals of S. If n<7 the product of G_{even} and the intersection of all ideals…

The purpose of this paper is to give a partial positive answer to a question raised by Khurana et al. as to whether a ring $R$ with stable range one and central units is commutative. We show that this is the case under any of the following…

Given an arbitrary integer $d>0$, we construct a homogeneous ideal $I$ of the polynomial ring $S = K[x_1, \ldots, x_{3d}]$ in $3d$ variables over a filed $K$ for which $S/I$ is a Cohen--Macaulay ring of dimension $d$ with the property that,…

We determine the metric dimension of the annihilating-ideal graph of a local finite commutative principal ring and a finite commutative principal ring with two maximal ideals. We also find the bounds for the metric dimension of the…

Let $I$ be a monomial ideal of the polynomial ring $S=K[x_1,...,x_4]$ over a field $K$. Then $S/I$ is sequentially Cohen-Macaulay if and only if $S/I$ is pretty clean. In particular, if $S/I$ is sequentially Cohen-Macaulay then $I$ is a…

It is proved that for a ring $R$ that is either an affine algebra over a field, or an equicharacteristic complete local ring, some power of the Jacobian ideal of $R$ annihilates $\mathrm{Ext}^{d+1}_{R}(-,-)$, where $d$ is the Krull…

Let $M$ be an $R$-module and $c$ the function from $M$ to the ideals of $R$ defined by $c(x) = \cap \lbrace I \colon I \text{is an ideal of} R \text{and} x \in IM \rbrace $. $M$ is said to be a content $R$-module if $x \in c(x)M $, for all…

The small finitistic dimension fPD$(R)$ of a ring $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we show that a commutative ring $R$ has fPD$(R)\leq d$ if and…

It is known that an ideal of a direct product of commutative unitary rings is directly decomposable into ideals of the corresponding factors. We show that this does not hold in general for commutative rings and we find necessary and…

In this article, all rings are commutative with nonzero identity. Let $M$ be an $R$-module. A proper submodule $N$ of $M$ is called a classical prime submodule, if for each $m\in M$ and elements $a,b\in R$, $abm\in N$ implies that $am\in N$…

Let $\varphi : S = k[y_0,..., y_n] \to R = k[y_0,...,y_n]$ be given by $y_i \to f_i$ where $f_0,...,f_n$ is an $R$-regular sequence of homogeneous elements of the same degree. A recent paper shows for ideals, $I_\Delta \subseteq S$, of…

Regarding polynomial functions on a subset $S$ of a non-commutative ring $R$, that is, functions induced by polynomials in $R[x]$ (whose variable commutes with the coefficients), we show connections between, on one hand, sets $S$ such that…

In this paper, new algebraic and topological results on purely-prime ideals of a commutative ring (pure spectrum) are obtained. Especially, Grothendieck type theorem is obtained which states that there is a canonical correspondence between…

We study integrality over rings (all commutative in this paper) and over ideal semifiltrations (a generalization of integrality over ideals). We begin by reproving classical results, such as a version of the "faithful module" criterion for…