فهرست مطالب
Journal of Algebraic Systems
Volume:12 Issue: 2, Winter-Spring 2025
- تاریخ انتشار: 1402/08/17
- تعداد عناوین: 12
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صفحات 193-209
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صفحات 211-235
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صفحات 237-256
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صفحات 257-267
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صفحات 269-282
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صفحات 283-299
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صفحات 327-346
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صفحات 347-366
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صفحات 367-377
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صفحات 379-390
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صفحات 391-401
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Pages 193-209
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.
Keywords: formal local cohomology, Local cohomology, Generalized -
Pages 211-235
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.
Keywords: prime filter, maximal filter, Super-max filter, Weakly Prime filter, Weak Top BL-algebra -
Pages 237-256
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.
Keywords: essential submodules, closed submodules, UC-module, Clique number, girth -
Pages 257-267
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 $k=2,3,4$.
Keywords: Multiplicative partition function, Set partitions, Partition function, Perfect square, Euler's Phi function -
Pages 269-282
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})$.
Keywords: Zigzag equations, Homotypical, Variety, Identity, Closed -
Pages 283-299
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.
Keywords: Minimal prime D-filter, π-filter, dense element, relatively complemented lattice, Hausdorff space -
Pages 301-326
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.
Keywords: Common neighborhood, spectrum, Energy, Conjugacy class graph -
Pages 327-346
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.
Keywords: $S$-act, Rees exact sequence, admissible exact sequence, admissibly projective acts -
Pages 347-366
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.
Keywords: generalized π-Baer ring, generalized π-Baer ∗-ring, generalized quasi-Baer ∗-ring, generalized Baer ring, generalized Baer ∗-ring -
Pages 367-377
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 $θ(Z(R))=Z(R)$.
Keywords: semiprime ring, centralizer, $, theta$-centralizer, epimorphism -
Pages 379-390
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.
Keywords: Monoform modules, Small monoform modules, Jacobson monoform modules -
Pages 391-401
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 $M$, $\omega(\Gamma_{\bot}(_R M))=\chi(\Gamma_{\bot}(_R M))$.
Keywords: chromatic number, clique number, finite graph, atomic module, semi-simple module