Shahrood University of TechnologyJournal of Algebraic Systems2345-512812220230101GENERALIZED FORMAL LOCAL COHOMOLOGY MODULES193209295310.22044/jas.2023.11630.1591ENShahram RezaeiDepartment of Mathematics, Payame Noor University (PNU), P.O. Box 19395-
4697, Tehran, Iran.Fatemeh LashkariDepartment of Mathematics, Payame Noor University (PNU), P.O. Box 19395-
4697, Tehran, Iran.Journal Article20220205Let $a$ be an ideal of a local ring $(R, m)$ and $M$ and $N$ two finitely generated <br /> $R$-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 <br /> $a$ 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.<br /><br />https://jas.shahroodut.ac.ir/article_2953_9518c73d9eeea8b76375a80db2c8f81a.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512812220230101WEAKLY PRIME AND SUPER-MAX FILTERS IN BL-ALGEBRAS211235295410.22044/jas.2023.12188.1638ENJavad MoghaderiDepartment of Mathematics, University of Hormozgan, P.O. Box 3995, Bandar
Abbas, Iran.Somayeh MotamedDepartment of Mathematics, Bandar Abbas Branch, Islamic Azad University,
Bandar Abbas, Iran.Journal Article20220804In 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.https://jas.shahroodut.ac.ir/article_2954_c91b4b626c982ea9aef911d2c5138a1a.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512812220230101A GRAPH ASSOCIATED TO ESPECIAL ESSENTIALITY OF SUBMODULES237256295610.22044/jas.2023.12356.1662ENMehdi Ebrahimi DorchehDepartment of Mathematics, Faculty of Mathematical Sciences, Malayer University, P.O. Box 65719-95863, Malayer, Iran.Saeid BagheriDepartment of Mathematics, Faculty of Mathematical Sciences, Malayer University, P.O. Box 65719-95863, Malayer, Iran.0000-0002-8394-1337Journal Article20221023Let $R$ be an associative ring with identity. In this paper we<br />associate 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 distinct<br />vertices $K$ and $L$ are considered to be adjacent if and only<br />if $K\cap L$ is an essential submodule of $K+L$.<br /><br />We investigate the relationship between some module theoretic<br />properties of $M$ such as minimality and closedness of<br />submodules with some graph theoretic properties of<br />$\Gamma_e(M)$. In general, this graph is not connected. We<br />study some special cases in which $\Gamma_e(M)$ is<br />complete or a union of complete connected components and give some examples illustrating each specific case.https://jas.shahroodut.ac.ir/article_2956_e3560fcd87507b8995c0316c56c6e70f.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512812220230101SOME RESULTS ON ORDERED AND UNORDERED FACTORIZATION OF A POSITIVE INTEGER257267295710.22044/jas.2023.12044.1618ENDaniel YaqubiDepartment of Computer science, University of Torbat e Jam, Torbat e Jam, Iran.Madjid MirzavaziriDepartment of Pure Mathematics, University of Ferdowsi, Mashhad, Iran.Journal Article20220629A 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 positive<br />integer $n$ such that each factor is at least $\ell$. In particular, by using elementary techniques, we give an explicit formula in cases where $k=2,3,4$.https://jas.shahroodut.ac.ir/article_2957_f43d94776f295f87221a2eb12437d419.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512812220230101ON CLOSED HOMOTYPICAL VARIETIES OF SEMIGROUPS-2269282295810.22044/jas.2023.12298.1656ENShabnam AbbasDepartment of Mathematics, Aligarh Muslim University, P.O. Box 202002, Aligarh,
India.Wajih AshrafDepartment of Mathematics, Aligarh Muslim University, P.O. Box 202002, Aligarh,
India.Rizwan AlamDepartment of Mathematics, Aligarh Muslim University, P.O. Box 202002, Aligarh,
India.Journal Article20220927In 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 $(n \in \mathbb{N})$.https://jas.shahroodut.ac.ir/article_2958_d574aba2c47e086615dafa1a645f9370.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512812220230101REGULAR FILTERS OF DISTRIBUTIVE LATTICES283299295910.22044/jas.2023.12585.1674ENSambasiva Rao MukkamalaDepartment of Mathematics, MVGR College of Engineering, Vizianagaram, P.O.
Box 535005, Andhra Pradesh, India.0000-0002-1627-9603Phaneendra Kumar AnanthapatnayakuniDepartment of Mathematics, MVGR College of Engineering, Vizianagaram, P.O. Box
535005, Andhra Pradesh, India.Journal Article20230110The 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.https://jas.shahroodut.ac.ir/article_2959_4a3f2fdf97b97fc8e6744c9d16df391f.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512812220230101COMMON NEIGHBOURHOOD SPECTRUM AND ENERGY OF COMMUTING CONJUGACY CLASS GRAPH301326296010.22044/jas.2023.12263.1650ENFirdous EeJannatDepartment of Mathematical Sciences, Tezpur University, Napaam-784028,
Sonitpur, India.Rajat KantiNathDepartment of Mathematical Sciences, Tezpur University, Napaam-784028,
Sonitpur, India.0000-0003-4766-6523Journal Article20220909In 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.https://jas.shahroodut.ac.ir/article_2960_fce7fe75318c80d71f3a0e7b2132414f.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512812220230101ADMISSIBLE (REES) EXACT SEQUENCES AND FLAT ACTS327346296110.22044/jas.2023.12249.1648ENElahe Nafarieh TalkhoonchehDepartment of Mathematics, Science and Research Branch, Islamic Azad
University, Tehran, Iran.Maryam SalimiDepartment of Mathematics, East Tehran Branch, Islamic Azad University, Tehran,
Iran.Hamid RasouliDepartment of Mathematics, Science and Research Branch, Islamic Azad
University, Tehran, Iran.Elham TavasoliDepartment of Mathematics, East Tehran Branch, Islamic Azad University, Tehran,
Iran.Abolfazl TehranianDepartment of Mathematics, Science and Research Branch, Islamic Azad
University, Tehran, Iran.Journal Article20220906Let $S$ be a commutative pointed monoid. <br />In this paper, some properties of admissible (Rees) <br />short exact sequences of $S$-acts are investigated. <br />In particular, it is shown that every admissible short exact sequence <br />of $S$-acts is Rees short exact. <br />In addition, a characterization of flat acts via preserving <br />admissible short exact sequences is established. <br />As a consequence, we show that for a flat $S$-act $F$, the functor <br />$F \otimes_{S} -$ preserves admissible morphisms. <br />Finally, it is proved that the class of flat $S$-acts is a subclass of <br />admissibly projective ones.https://jas.shahroodut.ac.ir/article_2961_70f8c4e27ec1516a6eb5d741a5ecad28.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512812220230101Generalized π-Baer *-rings347366296210.22044/jas.2023.12507.1670ENAli ShahidikiaDepartment of Mathematics, Dezful Branch, Islamic Azad University, Dezful, Iran.Haimd Haj Seyyed JavadiDepartment of Computer Engineering, Shahed University, Tehran, Iran.Journal Article20221213A *-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. <br />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.<br />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 compact<br />abelian 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.https://jas.shahroodut.ac.ir/article_2962_4184461fe62cfd5909d1b42360273a10.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512812220230101AN IDENTITY RELATED TO θ-CENTRALIZERS IN SEMIPRIME RINGS367377296310.22044/jas.2023.11856.1607ENAbbas Zivari-KazempourDepartment of Mathematics, Ayatollah Borujerdi University, Borujerd, Iran.Journal Article20220424Let $R$ be a $ 2$-torsion-free semiprime ring and $\theta$ be an epimorphism of $R$. In this paper, under special hypotheses, we prove that if $T: R\longrightarrow R$ <br />is an additive mapping such that<br />$$<br />T(xyx)=θ(x)T(y)θ(x),<br />$$<br />holds for all $x, y\in R$, then <br />$T$ is a $θ$-centralizer<br />either $R$ is unital or $θ(Z(R))=Z(R)$.https://jas.shahroodut.ac.ir/article_2963_c3bceba354de13e81079166621748469.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512812220230101JACOBSON MONOFORM MODULES379390296610.22044/jas.2023.12495.1668ENAbderrahim El MoussaouyDepartment of Mathematics, Faculty of Sciences, University of Mohammed First,
Oujda, Morocco0000-0001-9630-4698Journal Article20221210In this paper, we introduce and study the concept of Jacobson monoform modules which<br />is a proper generalization of that of monoform modules. We present a characterization of semisimple<br />rings in terms of Jacobson monoform modules by proving that a ring $R$<br />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.https://jas.shahroodut.ac.ir/article_2966_a7d278f0b4c68118ebfec4ccabcbae16.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512812220230101FURTHER STUDIES OF THE PERPENDICULAR GRAPHS OF MODULES391401296710.22044/jas.2023.11606.1587ENMaryam ShiraliDepartment of Mathematics, University of Yasouj, Yasouj, Iran.0000-0001-7190-9607Saeid SafaeeyanDepartment of Mathematics, University of Yasouj, Yasouj, Iran.Journal Article20220127In 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 <br />$B$ are called orthogonal, written $A\perp B$, if they do not have <br />non-zero isomorphic submodules. We associate a graph $\Gamma_{\bot}(M)$ to $M$ <br />with vertices <br />$\mathcal{M}_{\perp}=\{(0)\neq A\leq M\;|\; \exists (0)\neq B\leq M \; \mbox{such that}\; A\perp B\}$, <br />and for distinct $A,B\in <br />\mathcal{M}_{\perp}$, the vertices $A$ and $B$ are adjacent if and only if <br />$A\perp B$. The main object of this article is to study the <br />interplay of module-theoretic properties of $M$ with <br />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 $M$, $\omega(\Gamma_{\bot}(_R M))=\chi(\Gamma_{\bot}(_R M))$.https://jas.shahroodut.ac.ir/article_2967_2b8ca71a1ee38b671e412656cfa8516c.pdf