In this paper, we extend the concept of small subhypermodules to all types of hypermodules and give nontrivial examples for this concept. As an application, we define and study lifting hypermodules via small subhypermodules.

In this paper, we investigate left $phi$-biprojectivity of Segal algebras and abstract Segal algebras. We show that for some abstract Segal algebras with some mild conditions left $phi$-biprojectivity is equivalent with left $phi$-contractibility. Also, we characterize left $phi$-biprojectivity of a Segal algebra $S(G)$ in the terms of compactness of $G,$ where $G$ is a locally compact group. We introduce a class of abstract Segal algebras among Triangular Banach algebras. We show that some abstract Segal algebras related to triangular Banach algebras are not biprojective.

In this article we study and investigate the behavior of $r$-submodules (a proper submodule $N$ of an $R$-module $M$ in which $amin N$ with ${rm Ann}_M(a)=(0)$ implies that $min N$ for each $ain R$ and $min M$). We show that every simple submodule, direct summand, divisible submodule, torsion submodule and the socle of a module is an $r$-submodule and if $R$ is a domain, then the singular submodule is an $r$-submodule. We also introduce the concepts of $uz$-module (i.e., an $R$-module $M$ such that either ${rm Ann}_M(a)not=(0)$ or $aM=M$, for every $ain R$) and strongly $uz$-module (i.e., an $R$-module $M$ such that $aMsubseteq a^2M$, for every $ain R$) in the category of modules over commutative rings. We show that every Von Neumann regular module is a strongly $uz$-module and every Artinian $R$-module is a $uz$-module. It is observed that if $M$ is a faithful cyclic $R$-module, then $M$ is a $uz$-module if and only if every its cyclic submodule is an $r$-submodule. In addition, in this case, $R$ is a domain if and only if the only $r$-submodule of $M$ is zero submodule. Finally, we prove that $R$ is a $uz$-ring if and only if every faithful cyclic $R$-module is a $uz$-module.

Let $mathcal F_{mathcal P}( L)$ ($mathcal F_{mathcal P}^{*}( L)$) be the $f$-rings of all (bounded) frame maps from $mathcal P(mathbb R)$ to a frame $L$. $mathcal F_{{mathcal P}_{infty}}( L)$ is the family of all $fin mathcal F_{mathcal P}( L)$ such that ${uparrow}f(-frac 1n, frac 1n)$ is for any $ninmathbb N$ and the subring $mathcal F_{{mathcal P}_{K}}( L)$ is the family of all $fin mathcal F_{mathcal P}( L)$ such that ${{,mathrm{coz},}}(f)$ is compact. We introduce and study the concept of real ideals in $mathcal F_{mathcal P}( L)$ and $mathcal F_{mathcal P}^*( L)$. We show that every maximal ideal of $mathcal F_{mathcal P}^{*}( L)$ is real, and also we study the relation between the conditions ``$L$ is compact and ``every maximal ideal of $mathcal F_{mathcal P}(L)$ is real''. We prove that for every nonzero real Riesz map $varphi colon mathcal F_{mathcal P}( L)rightarrow mathbb R$, there is an element $p$ in $Sigma L$ such that $varphi=widetilde {p_{{{,mathrm{coz},}}}}$ if $L$ is a zero-dimensional frame for which $B(L)$ is a sub-$sigma$-frame of $L$ and every maximal ideal of $mathcal F_{mathcal P}( L)$ is real. We show that $mathcal F_{{mathcal P}_{infty}}(L)$ is equal to the intersection of all free maximal ideals of $ mathcal F_{mathcal P}^{*}(L) $ if $B(L)$ is a sub-$sigma$-frame of a zero-dimensional frame $L$ and also, $mathcal F_{{mathcal P}_{K}}(L)$ is equal to the intersection of all free ideals $mathcal F_{mathcal P}( L)$ (resp., $mathcal F_{mathcal P}^*( L)$) if $L$ is a zero-dimensional frame. Also, we study free ideals and fixed ideals of $mathcal F_{{mathcal P}_{infty}}( L)$ and $mathcal F_{{mathcal P}_{K}}( L)$.

Terms like commutativity degree, non-commuting graph and isoclinism are far well-known for much of the group theorists nowadays. There are so many papers about each of these concepts and also about their relationships in finite groups. Also, there are some recent researches about generalizing these notions in finite rings and their connexions.The concepts of commutativity degree and non-commuting graph are also extended to non-associative structures such as Moufang loops and some part of the known results in group theory in these contexts have been expanded to them.In this paper, we are going to generalize the notion of isoclinism in finite Moufang loops and then study the relationships between these three concepts. Among other results, we prove that two isoclinic finite Moufang loops have the same commutativity degree and if they have the same sizes of centers and commutants then they have isomorphic non-commuting graphs. Also, the converses of these results have been investigated.Furthermore, it has been proved that a finite simple group can be characterized by its non-commuting graph. We will prove the same is true for a finite simple Moufang loop by imposing one additional hypothesis, namely, the isoclinism of the regarding loops.

‎Let $G$ be a finite group‎. ‎We say that an element $g$ in $G$ is a vanishing element if there exists some irreducible character $chi$ of $G$ such that $chi(g)=0$‎. ‎In this paper‎, ‎we classify groups whose set of vanishing elements is exactly a conjugacy class‎.

In this paper, groups with trivial intersection between Frattini and derived subgroups are considered. First, some structural properties of these groups are given in an important special case. Then, some family invariants of each $n$-isoclinism family of such groups are stated. In particular, an explicit bound for the order of each center factor group in terms of the order of its derived subgroup is also provided.

The paper is devoted to concern a relationship between rough set theory and universal algebra. Notions of lower and upper rough approximations on an algebraic structure induced by an ideal are introduced and some of their properties are studied. Also, notions of rough subalgebras and rough ideals with respect to an ideal of an algebraic structure, which is an extended notion of subalgebras and ideals in an algebraic structure, are introduced and investigated.

A graph is called integral if all eigenvalues of its adjacency matrix are integers. Given a subset $S$ of a finite group $G$, the bi-Cayley graph $BCay(G,S)$ is a graph with vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid sin S, xin G}$. In this paper, we classify all finite groups admitting a connected cubic integral bi-Cayley graph.

Let $R$ be a commutative Noetherian ring and let $M$ be a finitely generated $R$-module. If $I$ is an ideal of $R$ generated by $M$-regular sequence, then we study the vanishing of the first $Tor$ functors. Moreover, for Artinian modules and coregular sequences we examine the vanishing of the first $Ext$ functors.