Journal of Algebraic Systems
Volume:12 Issue: 2, Winter-Spring 2025

Let $a$ be an ideal of a local ring $(R, m)$ and $M$ and  $N$ two finitely generated    $R$-modules. In this paper, we introduce the concept of generalized formal local cohomology modules. We define $i$-th generalized formal local cohomology module of $M$ and $N$ with respect to     $a$ by  $\mathfrak{F}_{a}^i(M,N) := \underset{n}{\varprojlim}H_m^i(M,N/{a}^{n}N )$ for $i\geq 0$. We prove several results concerning vanishing and finiteness properties of these modules.

‎In this paper‎, ‎the concepts of weakly prime filters and super-max filters in $\mathrm{BL}$-algebras are introduced‎, ‎and the relationships between them are discussed‎. ‎Also‎, ‎some properties and relations between these filters and other types of filters in $\mathrm{BL}$-algebras are given‎. ‎With some examples‎, ‎it is shown that these filters have differences‎. ‎After that‎, ‎the notions of weakly linear $\mathrm{BL}$-algebras and weak top $\mathrm{BL}$-algebras are defined and investigated‎. ‎Finally‎, ‎using the notion of a weakly prime filter‎, ‎a new topology on $\mathrm{BL}$-algebras is defined and studied‎.

‎Let $R$ be an associative ring with identity‎. ‎In this paper we‎‎associate to every $R$-module $M$ a simple graph $\Gamma_e(M)$‎, which we call it the essentiality graph of $M$. The vertices of $\Gamma_e(M)$ are nonzero submodules of $M$ and two distinct‎‎vertices $K$ and $L$ are considered to be adjacent if and only‎‎if $K\cap L$ is an essential submodule of $K+L$‎.‎‎We investigate the relationship between some module theoretic‎‎properties of $M$ such as minimality and closedness of‎‎submodules with some graph theoretic properties of‎‎$\Gamma_e(M)$‎. ‎In general‎, ‎this graph is not connected‎. ‎We‎‎study some special cases in which $\Gamma_e(M)$ is‎‎complete or a union of complete connected components and give some examples illustrating each specific case‎.

A well-known enumerative problem is to count the number of ways a positive integer $n$ can be factorised as $n=n_1\times n_2\times\cdots\times n_{k}$, where $n_1\geqslant n_2 \geqslant \cdots \geqslant n_{k} >1$. In this paper, we give some recursive formulas for the number of ordered/unordered factorizations of a positiveinteger $n$ such that each factor is at least $\ell$. In particular, by using elementary techniques, we give an explicit formula in cases where $k=2,3,4$.

In this paper we extended the results of paper\linebreak ``On Closed Homotypical Varieties of Semigroups" and have shown that the homotypical varieties of semigroups defined by the identities  $axy=x^nayx$$axy=xa^nya$[$axy=yay^nx$],$axy=xaya^n$[$axy=y^nayx$] and $axy=xayx^n$ are closed in itself, where $(n \in \mathbb{N})$.

The concepts of regular filters and π--filters are introduced in distributive lattices. A set of equivalent conditions is given for a D-filter to become a regular filter. For every D-filter, it is proved that there exists a homomorphism whose dense kernel is a regular filter. π--filters are characterized in terms of regular filters and congruences. Some equivalent conditions are given for the space of all prime π-filters to become a Hausdorff space.

In this paper, we compute the common neighbourhood (abbreviated as CN) spectrum and the common neighbourhood energy of commuting conjugacy class graph of several families of finite non-abelian groups. As a consequence of our results, we show that the commuting conjugacy class graphs of the groups $D_{2n}$, $T_{4n}$, $SD_{8n}$, $U_{(n,m)}$, $U_{6n}$, $V_{8n}$, $G(p, m, n)$ and some families of groups whose central quotient is isomorphic to $D_{2n}$ or $\mathbb{Z}_p \times \mathbb{Z}_p$, for some prime $p$, are CN-integral but not CN-hyperenergetic.

Let $S$ be a commutative pointed monoid. In this paper, some properties of admissible (Rees) short exact sequences of $S$-acts are investigated. In particular, it is shown that every admissible short exact sequence of $S$-acts is Rees short exact. In addition, a characterization of flat acts via preserving admissible short exact sequences is established. As a consequence, we show that for a flat $S$-act $F$, the functor $F \otimes_{S} -$ preserves admissible morphisms. Finally, it is proved that the class of flat $S$-acts is a subclass of admissibly projective ones.

‎A *-ring‎ ‎$R$ is called a generalized $\pi$-Baer *-ring‎, ‎if for any projection invariant left ideal $Y$ of $R$‎, ‎the right annihilator of $Y^n$‎ ‎is generated‎, ‎as a right ideal‎, ‎by a projection, ‎for some positive integer $n$‎, ‎depending on $Y$‎. ‎In this paper, ‎we‎ ‎study some properties of generalized $\pi$-Baer *-rings‎. ‎We show that this notion is well-behaved with respect to‎ ‎polynomial extensions, full matrix rings, and several classes of triangular matrix rings‎.‎ ‎We indicate interrelationships between the generalized $\pi$-Baer *-rings and related classes of rings such as‎ generalized ‎$\pi$-Baer rings‎, generalized ‎Baer *-rings‎, generalized quasi-Baer *-rings, and ‎$‎\pi$-Baer \s-rings. ‎We obtain algebraic examples which are generalized‎ $‎\pi$-Baer $ \ast $-rings but are not $‎\pi$-Baer *-rings‎. ‎We show that for pre-C*-algebras these two notions are equivalent‎.‎We obtain classes of Banach *-algebras‎ ‎which are generalized‎ $‎\pi$-Baer *-rings but are not $‎\pi$-Baer *-rings‎. We finish the paper by showing that for a locally compact‎‎abelian group $G$‎, ‎the group algebra $L^{1}(G)$ is a‎ ‎generalized $‎\pi$-Baer $*$-ring‎, ‎if and only if so is the group C*-algebra‎ ‎$C^{*}(G)$‎, ‎if and only if $G$ is finite‎.

‎Let $R$ be a $ 2$-torsion-free semiprime ring and $\theta$ be an epimorphism of $R$‎. ‎In this paper‎, ‎under special hypotheses‎, we prove that if $T‎: ‎R\longrightarrow R$‎ ‎is an additive mapping such that‎‎$‎‎$‎‎T(xyx)=θ(x)T(y)θ(x)‎,‎$‎‎$‎‎holds for all $x‎, ‎y\in R$‎, ‎then‎ ‎$T$ is a $θ$-centralizer‎either $R$ is unital‎ or $θ(Z(R))=Z(R)$.

In this paper, we introduce and study the concept of Jacobson monoform modules whichis a proper generalization of that of monoform modules. We present a characterization of semisimplerings in terms of Jacobson monoform modules by proving that a ring $R$is semisimple if and only if every $R$-module is Jacobson monoform. Moreover, we demonstrate that over a ring $R$, the properties monoform, Jacobson monoform, compressible, uniform and weakly co-Hopfian are all equivalent.

‎In this paper we continue our study of perpendicular graph of modules‎, ‎that was introduced in \cite{Hokkaido}‎. ‎Let $R$ be a ring and $M$ be an $R$-module‎. ‎Two modules $A$ and‎ ‎$B$ are called orthogonal‎, ‎written $A\perp B$‎, ‎if they do not have‎ ‎non-zero isomorphic submodules‎. ‎We associate a graph $\Gamma_{\bot}(M)$ to $M$‎ ‎with vertices‎ ‎$\mathcal{M}_{\perp}=\{(0)\neq A\leq M\;|\; \exists (0)\neq B\leq M \; \mbox{such that}\; A\perp B\}$‎, ‎and for distinct $A,B\in‎ ‎\mathcal{M}_{\perp}$‎, ‎the vertices $A$ and $B$ are adjacent if and only if‎ ‎$A\perp B$‎. ‎The main object of this article is to study the‎ ‎interplay of module-theoretic properties of $M$ with‎ ‎graph-theoretic properties of $\Gamma_{\bot}(M)$‎. ‎We study the clique number and chromatic number of $\Gamma_{\bot}( M)$‎. ‎We prove that if $\omega(\Gamma_{\bot}( M)) < \infty $ and $M$ has a simple submodule‎, ‎then $\chi(\Gamma_{\bot}(M)) < \infty $‎. ‎Among other results‎, ‎it is shown that for a semi-simple module $M$‎, ‎$\omega(\Gamma_{\bot}(_R M))=\chi(\Gamma_{\bot}(_R M))$‎.