Journal of Algebra and Related Topics
Volume:11 Issue: 2, Autumn 2023

‎Let $R$ be an associative ring with identity and $M$ an $R$-bimodule‎. ‎We introduce the generalization of $r$-clean rings called left $r$-clean $R$-bimodules‎, ‎defined without their endomorphism rings‎. ‎An $R$-bimodule $M$ is said to be left $r$-clean if each element is the sum of a left idempotent and a left regular element of $M$‎. ‎We present some properties of the left $r$-clean $R$-bimodule‎. ‎At the end of this paper‎, ‎we give the sufficient and necessary condition for an $R$-bimodule to form a left $r$-clean $R$-bimodule.‎

‎Let $R$ be a commutative Noetherian ring with identity‎, ‎$I$ be an ideal of $R$‎, ‎and $M$ be an $R$-module‎. ‎Let $k\geqslant‎ -‎1$ be an arbitrary integer‎. ‎In this paper‎, ‎we introduce the notions of $\Rad_M(I)$ and $\ara_M(I)$‎ ‎as the radical and the arithmetic rank of $I$ with respect to $M$‎, ‎respectively‎. ‎We show that the existence of some sort of regular sequences‎ ‎can be depended on $\dim M/IM$ and so‎, ‎we can get some information about local cohomology modules as well‎. ‎Indeed‎, ‎if $\ara_M(I)=n\geq 1$ and ${(\Supp_{R}(M/IM))}_{>k}=\emptyset$‎ ‎(e.g.‎, ‎if $\dim M/IM=k$)‎, ‎then there exist $n$ elements $x_1‎, ..., ‎x_n$ in $I$‎ ‎which is a poor $k$-regular $M$-sequence and generate an ideal‎ ‎with the same radical as $\Rad_M(I)$ and so‎ ‎$H^i_I(M)\cong H^i_{(x_1‎, ..., ‎x_n)}(M)$ for all $i\in \mathbb{N}_0$‎. ‎As an application‎, ‎we show that $\ara_M(I) \leq \dim M+1$‎, ‎which is a refinement of the inequality $\ara_R(I) \leq \dim R+1$ for modules‎, ‎attributed to Kronecker and Forster‎. ‎Then‎, ‎we prove‎ ‎$\dim M-\dim M/IM \leq \cd(I‎, ‎M) \leq \ara_M(I) \leq \dim M$‎, ‎if $(R‎, ‎\mathfrak{m})$ is a local ring and $IM \neq M$‎.

‎The set $\mathcal{C}_{c}(L)=\Big\{\alpha\in\mathcal{R}L‎ : ‎\big\vert\{ r\in\mathbb{R}‎ : ‎\coz(\alpha-{\bf r})\ne 1\big\}\big\vert\leq\aleph_0 \Big\}$ is a sub-$f$-ring of $\mathcal{R}L$‎, ‎that is‎, ‎the ring of all continuous real-valued functions on a completely regular frame $L$.‎ ‎The main purpose of this paper is to continue our investigation begun in \cite{a} of extending ring-theoretic properties in $\mathcal{R}L$ to‎ ‎the context of completely regular frames by replacing the ring $\mathcal{R}L$ with the ring $\mathcal{C}_{c}(L)$ to the context of zero-dimensional frames.‎ ‎We show that a frame $L$ is a $CP$-frame if and only if $\mathcal{C}_{c}(L)$ is a regular ring if and only if every ideal of $\mathcal{C}_{c}(L)$ is pure if and only if $\mathcal{C}_c(L)$ is an Artin-Rees ring if and only if every ideal of $\mathcal{C}_c(L)$ with the Artin-Rees property is an Artin-Rees ideal if and only if the factor ring $\mathcal{C}_{c}(L)/\langle\alpha\rangle$ is an Artin-Rees ring for any $\alpha\in\mathcal{C}_{c}(L)$ if and only if every minimal prime ideal of $\mathcal{C}_c(L)$ is an Artin-Rees ideal.‎

‎In this paper‎, ‎our aim is to introduce and study the essential submodules of an $R$-module $M$ relative to an arbitrary submodule $T$ of $M$‎. ‎Let $T$ be an arbitrary submodule of an $R$-module $M$‎, ‎then we say that a submodule $N$ of $M$ is an essential submodule of $M$ relative to $T$‎, ‎whenever for every submodule $X$ of $M$‎, ‎$N\cap X\subseteq T$ implies that‎ ‎$(T:M)\subseteq ^{e}{\rm Ann}(X)$‎. ‎We investigate some new results concerning to this class of submodules‎. ‎Among various results we prove that for a faithful multiplication $R$-module $M$‎, ‎if the submodule $N$ of $M$ is an essential submodule of $M$ relative to $T$‎, ‎then $(N:M)$ is an essential ideal of $R$ relative to $(T:M)$‎. ‎The converse is true if $M$ is moreover a finitely generated module‎.

‎We introduce the concept of weakly nil‎ - ‎semicommutative or WNSC rings and provide a condition that establishes the equivalence of WNSC rings to three generalised classes of semicommutative rings‎. ‎We prove the equivalence between WNSC Laurent polynomial rings and WNSC polynomial rings‎. ‎We supply examples of these classes of rings by considering Nagata and Dorroh extensions‎. ‎We also give a characterization for a ring of Morita context with zero pairings to be WNSC‎.

$EL$-hypergroups were defined by Chvalina 1995. Tillnow, no exact statistics of $EL$-hypergroups have been done.\\ Moreover, there is no classification of $EL$-hypergroups and $EL^2$-hypergroups even over small sets.In this paper, we classify all $EL$-(semi)hypergroups over sets withtwo elements obtained from quasi ordered semigroups. Also, we characterize all quasi ordered $H_v$-group and then we enumerate the number of $EL^2$-$H_v$-hypergroups and $EL^2$-hypergroups of order 2.

Let $L$ be a finite dimensional Lie algebra over an arbitrary field $F$. In this paper, we prove that the class of finite nilpotent(solvable) Lie algebras is an example of formation. Furthermore, we conclude that every finite Lie algebra has a nilpotent(solvable) residual. Finally we prove some results on Frattini and Fitting subalgebras of the nilpotent Lie algebra $L$.

In this paper, we study the transfer of some algebraic properties from the ring $R$ to the ring of skew Hurwitz series $(HR, \omega)$, where $\omega$ is an automorphism of $R$ and vice versa. Different properties of skew Hurwitz series are studied with respect to various clean ring structures and semiclean ring structures.

‎Assuming that $\mathcal{R}$ is an associative ring with prime ideal $P$‎, ‎this paper investigates the commutativity of the quotient ring $\mathcal{R}/P$‎, ‎as well as the possible forms of generalized derivations satisfying certain algebraic identities on $\mathcal{R}.$ We give results on strong commutativity‎, ‎preserving generalized derivations of prime rings‎, ‎using our theorems‎. ‎Finally‎, ‎an example is given to show that the restrictions on the ideal $P$ are not superfluous.‎

‎We introduce the notion of lifting Baer modules‎, ‎as a generalization of both Baer and lifting modules and give some of their properties‎. ‎A module $M$ is called lifting Baer if right annihilator of a left ideal of ${\rm End}(M)$ lies above a direct summand of M‎. ‎Also‎, ‎we define the concepts of $r$-supplemented and amply $r$-supplemented modules‎. ‎It is shown that an amply $r$-supplemened module M that every supplement submodule‎, ‎is a direct summand of $M$‎, ‎is lifting Baer‎. ‎The relationships between Baer modules and lifting Baer modules are investigated.‎ ‎Morever‎, ‎we prove that the endomorphism ring of any lifting Baer module is‎ ‎lifting Baer ring.‎

‎Let $\Gamma$ be a finite group and $S$ be a non-empty subset of $\Gamma$‎. ‎A Cayley graph of the group $\Gamma$‎, ‎denoted by $Cay(\Gamma‎, ‎S)$ is defined as a simple graph that its vertices are the elements of $\Gamma$ and two vertices $u$ and $v$ are adjacent if $uv^{-1} \in \Gamma$. ‎The minimum edge dominating energy of Cayley graph $Cay(\Gamma‎, ‎S)$ is equal to the sum of the absolute values of eigenvalues of the minimum edge dominating matrix of graph $Cay(\Gamma‎, ‎S)$‎. ‎In this paper‎, ‎we estimate the minimum edge dominating energy of the Cayley graphs for the finite group $S_n$‎.

The concepts of Dokdo sub-hoops and Dokdo filters are introduced‎, ‎and their properties are investigated.‎‎ The relationship between Dokdo sub-hoops and Dokdo filters is discussed‎, ‎and‎‎ characterizations of Dokdo sub-hoops and Dokdo filters are established.‎