Journal of Algebra and Related Topics
Volume:11 Issue: 1, Spring 2023

The notion of (closed) \L{}ukasiewicz fuzzy ideal is introduced, and several properties are investigated.The relationship between \L{}ukasiewicz fuzzy subalgebra and \L{}ukasiewiczfuzzy ideal is discussed, and characterization of a \L{}ukasiewicz fuzzy ideal is considered.Conditions for a \L{}ukasiewicz fuzzy subalgebra to be a \L{}ukasiewicz fuzzy ideal are provided, andconditions for the $\in$-set, $q$-set and $O$-set to be ideals are explored.

Let $(G,M)$ be a pair of groups, in which $M$ is a normal subgroup of a group $G.$ We study some properties of $n$-isoclinism of pair of groups. In fact, we show that the subgroups and quotient groups of two $n$-isoclinism pair of groups are $m$-isoclinic for all $m\le n.$ Moreover, the properties of $\pi$-pair and supersolvable pair of groups which are invariant under $n$-isoclinism has be studied.

‎‎Let $R$ be a commutative ring with non-zero identity, $S\subseteq R$ be a multiplicatively closed subset of $R$ and let $M$ be an $R$-module.  A submodule $N$ of $M$ with $(N:_{R}M)\cap S=\emptyset$ is said to be almost $S$-prime, if there exists an $s\in S$ such that whenever $rm\in N-(N:_{R}M)N$, then $sm\in N$ or $sr\in (N:_{R}M)$ for each $r\in R$, $m\in M$. The aim of this paper is to introduce and investigate some properties of the notion of almost $S$-prime submodules, especially in multiplication modules. Moreover, we investigate the behaviour of this structure under module homomorphisms, localizations, quotient modules, Cartesian product. Finally, we state two kinds of submodules of the amalgamation module along an ideal and investigate conditions under which they are almost $S$-prime.

We show that the radical bundle of an algebra bundle is a characteristic ideal bundle. Further we prove an algebra bundle is semisimple if and only if its derivation algebra bundle is either semisimple or zero.

Let $R$ be a commutative ring with identity, and let $M$ be an $R$-module.  The Cohen's theorem is the classic result that a ring is Noetherian if and only if its prime ideals are finitely generated. Parkash and Kour obtained a new version of Cohen's theorem for modules, which states that a finitely generated $R$-module $M$ is Noetherian if and only if for every prime ideal $p$ of $R$ with $Ann(M) \subseteq p$, there exists a finitely generated submodule $N$ of $M$ such that $pM \subseteq N \subseteq M(p)$, where $M(p) = \{x \in M | sx \in pM \,\,\textit{for some} \,\, s \in R \backslash p\}$. In this paper, we prove this result for some classes of modules.

In this paper, we enumerate the parameter matrices of all perfect 2-colorings of the generalized prism graph Cn × C3, where n ≥ 3,. We also present some generalized results for Cn × Cm, where m, n ≥ 3.

The concept of pre-open set in topological spaces was introduced by Mashhour \cite{ma} and the concept of BCK-algebras was defined by Iseki \cite{ise}. The aim of this paper is to supply the BCK-algebra by a topology that makes the operation defined on $X$ satisfy a special type of continuity with help of pre-open sets we call it p-topological BCK-algebra. This concept is an extension of the concept of topological BCK-algebra which was defined in \cite{dong}. By using properties of pre-open sets and axioms of BCK-algebras, several topological properties on BCK-algebras are found.

In this paper, we focus on the concept of locally $\kappa$-presented representations of quiver and we introduce two classes of objects of representations of the quiver on certain Grothendieck category related to this concept which forms a complete cotorsion pair.

‎Let $G$ be a group with identity $e$ ‎and ‎‎$‎R$ be a multiplicative ‎hyperring‎‎. ‎W‎e introduce and study the notions of graded $2$-absorbing and graded $2$-absorbing primary hyperideals of a graded multiplicative hyperring $R$ which are generalizations of prime ‎hyperideals‎‎. ‎We present basic properties and characterizations of these graded hyperideals and homogeneous components‎. Among various results we prove that ‎the intersection of ‎‎‎two ‎graded‎ prime ‎hyperideals ‎is a‎ ‎graded ‎‎$‎2‎‎$‎-absorbing ‎hyperideal.‎

Our intention in this paper is to prove the following. Let R be a ring with an idempotent element (0, 1 ̸=)e and f be a multiplicative b-generalized derivation on R. Then we show that f is additive byimposing certain conditions on the ring R.

After introducing the concept of single valued neutrosophic ideal in a pseudo MV-algebra, its properties are examined. The various conditions under which a single valued neutrosophic set can be a single valued neutrosophic ideal are examined. Characterizations of a single valued neutrosophic ideal of a pseudo MV-algebra are considered.

In this paper, we investigate the relationship between symmetric bi-semiderivations and Lie ideals of a prime ring. Additionally, we extend some well-known results concerning symmetric biderivations of prime rings to symmetric bi-semiderivations.