*Algebraic structures and their applications*,
Volume 7, Issue 1, Pages 1-10

DOI:
10.29252/as.2020.1615

The first part of the paper is concerned to relationship between the sets of associated primes of the generalized $d$-local cohomology modules and the ordinary generalized local cohomology modules. Assume that $R$ is a commutative Noetherian local ring, $M$ and $N$ are finitely generated $R$-modules and $d, t$ are two integers. We prove that $Ass H^t_d(M,N)=bigcup_{Iin Phi} Ass H^t_I(M,N)$ whenever $H^i_d(M,N)=0$ for all $i t$ and $Phi={I: I text{ is an ideal of} R text{ with} dim R/Ileq d }$. In the second part of the paper, we give some information about the non-vanishing of the generalized $d$-local cohomology modules. To be more precise, we prove that $H^i_d(M,R)neq 0$ if and only if $i=n-d$ whenever $R$ is a Gorenstein ring of dimension $n$ and $pd_R(M)infty$. This result leads to an example which shows that $Ass H^{n-d}_d(M,R)$ is not necessarily a finite set.

*Algebraic structures and their applications*,
Volume 7, Issue 1, Pages 11-20

DOI:
10.29252/as.2020.1620

We will study modules whose cofinite submodules have weak generalized-$delta$-supplements. We attempt to investigate some properties of cofinitely weak generalized $delta$-supplemented modules. We will prove for a module $M$ and a semi-$delta$-hollow submodule $N$ of $M$ that, $M$ is cofinitely weak generalized $delta$-supplemented if and only if $frac{M}{N}$ is cofinitely weak generalized $delta$-supplemented. Also we show that any $M$-generated module is cofinitely weak generalized $delta$-supplemented module, where $M$ is cofinitely weak generalized $delta$-supplemented. We obtain some other results about this kind of modules. We will study modules whose cofinite submodules have weak generalized-$delta$-supplements. We attempt to investigate some properties of cofinitely weak generalized $delta$-supplemented modules. We will prove for a module $M$ and a semi-$delta$-hollow submodule $N$ of $M$ that, $M$ is cofinitely weak generalized $delta$-supplemented if and only if $frac{M}{N}$ is cofinitely weak generalized $delta$-supplemented. Also we show that any $M$-generated module is cofinitely weak generalized $delta$-supplemented module, where $M$ is cofinitely weak generalized $delta$-supplemented. We obtain some other results about this kind of modules.

*Algebraic structures and their applications*,
Volume 7, Issue 1, Pages 21-28

DOI:
10.29252/as.2020.1621

Let $(R,mathfrak{m})$ be a Noetherian local ring, $M$ a finitely generated $R$-module, and $mathfrak{a}$ an ideal of $R$. We define the $mathfrak{a}$-minimum dimension $d(mathfrak{a},M)$ of $M$ by $$d(mathfrak{a},M)=Min{dim frac{R}{mathfrak{p}+mathfrak{a}}:mathfrak{p}in Assh_{R}(M)}.$$ In this paper, we show that $cd(mathfrak{a},M)geq dim M-d(mathfrak{a},M)$ and we give some sufficient conditions and characterization for the equality to hold true.

*Algebraic structures and their applications*,
Volume 7, Issue 1, Pages 29-40

DOI:
10.29252/as.2020.1683

Cofiniteness of the generalized local cohomology modules $H^{i}_{mathfrak{a}}(M,N)$ of two $R$-modules $M$ and $N$ withrespect to an ideal $mathfrak{a}$ is studied for some $i^{,}s$ witha specified property. Furthermore, Artinianness of$H^{j}_{mathfrak{b}_{0}}(H_{mathfrak{a}}^{i}(M,N))$ isinvestigated by using the above result, in certain graded situations, where $mathfrak{b}_{0}$ is an ideal of $R_{0}$ and$mathfrak{a}=mathfrak{a}_{0}+R_{+}$ such that$mathfrak{b}_{0}+mathfrak{a}_{0}$ is an $mathfrak{m}_{0}$-primary ideal.

*Algebraic structures and their applications*,
Volume 7, Issue 1, Pages 41-47

DOI:
10.29252/as.2020.1686

In this paper, it is shown that $ (mathcal{V}, mathfrak{X}) $ is a Schur pair if and only if the Baer-invariant of an $mathfrak{X}$-group with respect to $ mathcal{V}$ is an $mathfrak{X}$-group. Also, it is proved that a locally $mathfrak{X}$ class inherited the Schur pair property of , whenever $mathfrak{X}$ is closed with respect to forming subgroup, images and extensions of its members. Subsequently, many interesting predicates about some generalizations of Schur's theorem and Schur multiplier of groups will be concluded.

*Algebraic structures and their applications*,
Volume 7, Issue 1, Pages 49-57

DOI:
10.29252/as.2020.1717

Let $M$ be a module over a ring $R$. We call $M$,$delta$-$H$-supplemented provided for every submodule $N$ of $M$ there is a direct summand $D$ of $M$ such that $M=N+X$ if and only if $M=D+X$ for every submodule $X$ of $M$ with $M/X$ singular. We prove that $M$ is $delta$-$H$-supplemented if and only if for every submodule $N$ of $M$ there exists a direct summand $D$ of $M$ such that $(N+D)/Nll_{delta} M/N$ and $(N+D)/Dll_{delta} M/D$.

*Algebraic structures and their applications*,
Volume 7, Issue 1, Pages 59-67

DOI:
10.29252/as.2020.1718

In this paper, we use the primitive permutation representations of the simple groups $PSL_2(53)$, $PSL_2(61)$ and $PSL_2(64)$ and construct 1-designs by the Key-Moori Method 1.It is shown that the groups $PSL_2(53)$, $PSL_2(53)text{:}2$, $PSL_2(61)$, $PSL_2(61)text{:}2$, $PSL_2(64)$, $PSL_2(64)text{:}2$, $PSL_2(64)text{:}3$ and $PSL_2(64)text{:}6$ appear as the full automorphism groups of these obtained designs.

*Algebraic structures and their applications*,
Volume 7, Issue 1, Pages 69-82

DOI:
10.29252/as.2020.1719

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.

*Algebraic structures and their applications*,
Volume 7, Issue 1, Pages 83-99

DOI:
10.29252/as.2020.1720

In this paper, we define the notion of strongly annihilating-submodule graph of modules. This graph is a straightforward common generalization of the annihilating-submodule graph and the annihilating-ideal graph. In addition to providing the properties of this graph in general, we investigate the behavior of the graph when modules are reduced or divisible.