21,274 Results Found

A BN-algebra is a non-empty set $X$ with a binary operation “$\ast$” and a constant 0 that satisfies the following axioms: $\left(B1\right) x\ast x=0,$ $\left(B2\right) x\ast 0=x,$ and $\left(BN\right) \left(x\ast y\right)\ast z=\left(0\ast z\right)\ast (y\&low$...

We consider an operator $G_{i,I}$ generated by a given interior operator i and an ideal I. Among the properties of the operator $G_{i,I}$, two properties denoted by ${\mathcal{P}}_i$ and ${\mathcal{Q}}_i$ are remarkable. The ideals with the property ${\mathcal{P}}_i$ are called dominated ideals, and the...

In this paper, we introduce the concepts of almost right primary ideals and almost nilary ideals and study their related results. We compare almost right primary ideals with other types of ideals, such as right primary ideals and weakly right primary...

The purpose of this paper is to introduce the concept of graded 2-prime ideals as a new generalization of graded prime ideals. We show that graded 2-prime ideals and graded semi-prime ideals are different. Furthermore, we show that graded 2-prime ide...

We describe first-degree prime ideals of biquadratic extensions in terms of the first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms. The c...

Let R be a commutative graded ring with unity, S be a multiplicative subset of homogeneous elements of R and P be a graded ideal of R such that $P\bigcap S=\varnothing$. In this article, we introduce the concept of graded S-primary ideals which is a generalization of g...

Group actions are a valuable tool for investigating the symmetry and automorphism features of rings. The concept of fuzzy ideals in rings has been expanded with the introduction of fuzzy primary, weak primary, and semiprimary ideals. This paper explo...

Prime ideals and their generalizations are crucial in numerous research areas, particularly in commutative algebra. The concept of generalization of prime ideals begins with the study of weakly prime ideals. Since then, subsequent works aimed at expa...

Neutrosophy is a recent section of philosophy. It was initiated in 1980 by Smarandache. It was presented as the study of origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. In this paper, we int...

Given a homogeneous ideal $I\subseteq k[x_0,\dots ,x_n]$, the Containment problem studies the relation between symbolic and regular powers of I, that is, it asks for which pairs $m,r\in \mathbb{N}$, $I^{\left(m\right)}\subseteq I^r$ holds. In the last years, several conjectures have be...

In this paper, we acquaint new kinds of ideals of BCK-algebras built on tripolar picture fuzzy structures. In fact, the notions of tripolar picture fuzzy ideal, tripolar picture fuzzy implicative ideal (commutative ideal) of BCK-algebra are introduce...

In 1982, Stanley predicted a combinatorial upper bound for the depth of any finitely generated multigraded module over a polynomial ring. The predicted invariant is now called the Stanley depth. Duval et al. found a counterexample for Stanley’s...

Let $k$ be an algebraically closed field of characteristic zero, and $R/I$ and $S/J$ be algebras over $k$ . ${\Omega }_1\left(R/I\right)$ and ${\Omega }_1\left(S/J\right)$ denote universal module of first order derivation over $$...

There is a correspondence between equivalence classes of fuzzy ideals, on a commutative ring, and decreasing gradual ideals. In this paper, we explore how to construct a fuzzy ideal starting from any decreasing gradual ideal $\sigma$. To achieve this,...

In this paper, the notion of cubic intuitionistic q-ideals in $BCI$ -algebras is introduced. A relationship between a cubic intuitionistic subalgebra, a cubic intuitionistic ideal, and a cubic intuitionistic q-ideal is discussed. Conditions fo...

The notion of (normal) m-polar $(\in ,\in )$ -fuzzy p-ideals of BCI-algebras is introduced, and several properties are investigated. Relations between an m-polar $(\in ,\in )$ -fuzzy ideal and an m-polar $(\in ,\&isin$...

The concept of a commutative generalized neutrosophic ideal in a $BCK$ -algebra is proposed, and related properties are proved. Characterizations of a commutative generalized neutrosophic ideal are considered. Also, some equivalence relations...

In this paper, we give a new criterion for the Cohen–Macaulayness of vertex splittable ideals, a family of monomial ideals recently introduced by Moradi and Khosh-Ahang. Our result relies on a Betti splitting of the ideal and provides an induct...

The study of symmetry is one of the most important and beautiful themes uniting various areas of contemporary arithmetic. Algebraic structures are useful structures in pure mathematics for learning a geometrical object’s symmetries. In order to...

We study the concept of i-ideal of an ordered n-ary semigroup and give a construction of the i-ideal of an ordered n-ary semigroup generated by its nonempty subset. Moreover, we study the notions of prime, weakly prime, semiprime and weakly semiprime...

The notion of a neutrosophic commutative N -ideal in BCK-algebras is introduced, and several properties are investigated. Relations between a neutrosophic N-ideal and a neutrosophic commutative N-ideal are discussed. Characterizations of a neutrosoph...

Recently, fuzzy multisets have come to the forefront of scientists’ interest and have been used for algebraic structures such as groups, rings, and near rings. In this paper, we first summarize the knowledge about algebraic structure of fuzzy multise...

All rings considered are commutative with identity, and all modules are assumed to be unital. In this paper, we study R-modules in which every quasi-primary submodule is also primary; we refer to such modules as satisfying condition (*). We present s...

In this article, we define and study graded 1-absorbing prime ideals and graded weakly 1-absorbing prime ideals in non-commutative graded rings as a new class of graded ideals that lies between graded prime ideals (graded weakly prime ideals) and gra...

The study of symmetry is one of the most important and beautiful themes uniting various areas of contemporary arithmetic. Algebraic structures are useful structures in pure mathematics for learning a geometrical object’s symmetries. In this pap...

The notion of a neutrosophic positive implicative $\mathcal{N}$ -ideal in $BCK$ -algebras is introduced, and several properties are investigated. Relations between a neutrosophic $\mathcal{N}$ -ideal and a neutrosophic positive implicative $\mathcal{N}$ -ideal are...

In this article, we introduce and examine the concept of graded weakly strongly quasi primary ideals. A proper graded ideal P of R is said to be a graded weakly strongly quasi primary (shortly, Gwsq-primary) ideal if whenever $0\ne xy\in P$, for some...

Let $\mathbb{S}$ be a numerical monoid, while a ${\mathcal{P}}_{\mathrm{fin}}\left(\mathbb{S}\right)$-monoid S is a monoid generated by a finite number of finite non-empty subsets of $\mathbb{S}$. That is, S is a non-cancellative commutative monoid obtained from the sumset of finite non-negative integer sets. This wo...

Since bipolar quantum linear algebra (BQLA), under two operations–-addition and multiplication—demonstrates the properties of semirings, and since ideals play an important role in abstract algebra, our results are compelling for the ideal...

This article presents some properties of a special class of interior operators generated by ideals. The mathematical framework is given by complete domains, namely complete posets in which the set of minimal elements is a basis. The first part of the...

The notion of hybrid ideals in $BCK/BCI$ -algebras is introduced, and related properties are investigated. Characterizations of hybrid ideals are discussed. Relations between hybrid ideals and hybrid subalgebras are considered. Characteriz...

The notions of hyperfuzzy ideals in $BCK/BCI$ -algebras are introduced, and related properties are investigated. Characterizations of hyperfuzzy ideals are established. Relations between hyperfuzzy ideals and hyperfuzzy subalgebras are dis...

The aim of this study is to provide a generalization of prime vague $\Gamma$-ideals in $\Gamma$-rings by introducing non-symmetric 2-absorbing vague weakly complete $\Gamma$-ideals of commutative $\Gamma$-rings. A novel algebraic structure of a primary v...

Let K be a field and let $S=K[x_1,\cdots ,x_n]$ be a polynomial ring over K. We analyze the extremal Betti numbers of special squarefree monomial ideals of S known as the t-spread strongly stable ideals, where t is an integer $\ge 1$...

Our main objective is to introduce the innovative concept of $(\alpha ,\beta )$ -bipolar fuzzy ideals and $\left(\alpha ,\beta \right)$ -bipolar fuzzy generalized bi-ideals in ordered ternary semigroups by using the idea of belongingness a...

In 2020, Kang et al. introduced the concept of a multipolar intuitionistic fuzzy set of finite degree, which is a generalization of a k-polar fuzzy set, and applied it to a BCK/BCI-algebra. The specific purpose of this study was to apply the concept...

As a generalization of a neutrosophic set, the notion of MBJ-neutrosophic sets is introduced by Mohseni Takallo, Borzooei and Jun, and it is applied to BCK/BCI-algebras. In this article, MBJ-neutrosophic set is used to study commutative ideal in BCI-...

In this paper we study some geometric properties of the algebraic set associated to the binomial edge ideal of a graph. We study the singularity and smoothness of the algebraic set associated to the binomial edge ideal of a graph. Some of these algeb...

In the study of algebraic structures related to logical systems, ideals and filters have different meanings and are algebraic notions related to logical provable formulas. Unlike the classical Boolean lattice theory, ideals and filters are not dual n...

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then, we show that for the edge ideal $I\left(G\right)$ of a graph G, the induced matching number of G is an upper bound for the v-number of $I\left(G\right)$ when G is very well-cov...

Let G be a group and R be a G-graded ring. In this paper, we present and examine the concept of graded weakly 2-absorbing ideals as in generality of graded weakly prime ideals in a graded ring which is not commutative, and demonstrates that the symme...

Based on the definitions of fuzzy associative algebras and fuzzy ideals, it is proven that the intersections of fuzzy subalgebras are fuzzy subalgebras, and the intersections of fuzzy ideals are fuzzy ideals. Moreover, we prove that the $\mathrm{kernels}$ of fu...

Multi-polar vagueness in data plays a prominent role in several areas of the sciences. In recent years, the thought of m-polar fuzzy sets has captured the attention of numerous analysts, and research in this area has escalated in the past four years....

The concepts of a positive implicative ( $\in ,$ ∈)-intuitionistic fuzzy ideal and a positive implicative falling intuitionistic fuzzy ideal are introduced, and several properties are investigated. Characterizations of a positive implica...

The concept of fuzzy multiset is well established in dealing with many real life problems. It is possible to find various applications of algebraic hypercompositional structures in natural, technical and social sciences, where symmetry, or the lack o...

Let $a,b$ and $n>1$ be three positive integers such that a and ${\sum }_{j=0}^{n-1}{b}^j$ are relatively prime. In this paper, we prove that the toric ideal I associated to the submonoid of $\mathbb{N}$ generated by $\left\{{\sum }_{j=0}^{n-1}{b}^j\right\}\cup \{{\sum }_{j=0}^{n-1}{b}^j+a{\sum }_{}^{}$...

In computer programming languages, partial additive semantics are used. Since partial functions under disjoint-domain sums and functional composition do not constitute a field, linear algebra cannot be applied. A partial ring can be viewed as an alge...

In this paper, we generalize and study the concept of Hadamard product of ideals of projective varieties to the case of monomial ideals. We have a research direction similar to the one of the join of monomial ideals contained in a paper of Sturmfels...

In the present paper, we study the normality of the toric rings of stable set polytopes, generators of toric ideals of stable set polytopes, and their Gröbner bases via the notion of edge polytopes of finite nonsimple graphs and the results on t...

The notion of anti-intuitionistic fuzzy soft a-ideals of $BCI$ -algebras is introduced and several related properties are investigated. Furthermore, the operations, namely; AND, extended intersection, restricted intersection, and union on anti...