Journal of Algebraic Systems
Volume:1 Issue: 1, Summer - Autumn 2013

Let $R$ be a commutative Noetherian ring with non-zero identity and $fa$ an ideal of $R$. Let $M$ be a finite $R$--module of finite projective dimension and $N$ an arbitrary finite $R$--module. We characterize the membership of the generalized local cohomology modules $lc^{i}_{fa}(M,N)$ in certain Serre subcategories of the category of modules from upper bounds. We define and study the properties of a generalization of cohomological dimension of generalized local cohomology modules. Let $mathcal S$ be a Serre subcategory of the category of $R$--modules and $n > pd M$ be an integer such that $lc^{i}_{fa}(M,N)$ belongs to $mathcal S$ for all $i> n$. Then, for any ideal $fbsupseteq fa$, it is also shown that the module $lc^{n}_{fa}(M,N)/{fb}lc^{n}_{fa}(M,N)$ belongs to $mathcal S$.

Recently, the algebraic theory of MV -algebras is intensively studied. In this paper, we extend the concept of derivation of $MV$-algebras and we give some illustrative examples. Moreover, as a generalization of derivations of $MV$ -algebras we introduce the notion of $f$-derivations and $(f; g)$-derivations of $MV$-algebras. Also, we investigate some properties of them.

Nets, useful topological tools, used to generalize certain concepts that may only be general enough in the context of metric spaces. In this work we introduce this concept in an $S$-poset, a poset with an action of a posemigroup $S$ on it which is a very useful structure in computer sciences and interesting for mathematicians, and give the the concept of $S$-net. Using $S$-nets and its convergency we also give some characterizations of separated $S$-posets. Also, introducing the net-closure operators, we investigate the counterparts of topological separation axioms on $S$-posets and study their relation to separated $S$-posets.

In this paper, we verify the solvability of free product of finite cyclic groups with topological methods. We use Cayley graphs and Everitt methods to construct suitable 2 complexes corresponding to the presentations of groups and their commutator subgroups. In particular, using these methods, we prove that the commutator subgroup of $mathbb{Z}_{m}*mathbb{Z}_{n}$ is free of rank $(m-1)(n-1)$ for all $m,ngeq2$.

In this paper we consider Selberg-type square matrices integrals with focus on Kummer beta types I & II integrals. For generality of the results for real normed division algebras, the generalized matrix variate Kummer-beta types I & II are defined under the abstract algebra. Then Selberg-type integrals are calculated under orthogonal transformations.

In this paper we introduce the notions of G∗L-module and G∗L-module which are two proper generalizations of δ-lifting modules. We give some characteriza tions and properties of these modules. We show that a G∗L-module decomposes into a semisimple submodule M1 and a submodule M2 of M such that every non-zero submodule of M2 contains a non-zero δ-cosingular submodule.