Journal of Algebraic Systems
Volume:6 Issue: 2, Winter - Spring 2019

A permutation with no fixed points is called a derangement. The subset D of a permutation group is derangement if all elements of D are derangement. Let G be a permutation group, a derangement graph is one with vertex set G and derangement set D as connecting set. In this paper, we determine the spectrum of derangement graphs of order a product of three primes.

We introduce and study the concept of α -semi short modules. Using this concept we extend some of the basic results of α-short modules to α-semi short modules. We observe that if M is an α-semi short module then the dual perfect dimension of M is α or α+1. %In particular, if a semiprime ring R is α-semi short as an R-module, then its Noetherian dimension either is α or α+1.

In this paper, first we study the semi maximal filters in linear BL-algebras and we prove that any semi maximal filter is a primary filter. Then, we investigate the radical of semi maximal filters in BL-algebras. Moreover, we determine the relationship between this filters and other types of filters in BL-algebras and G\"{o} del algebra. Specially, we prove that in a G\"{o}del algebra, any fantastic filter is a semi maximal filter and any semi maximal filter is an (n-fold) positive implicative filter. Also, in a BL -algebra, any semi maximal and implicative filter is a positive implicative filter. Finally, we give an answer to the open problem in [S. Motamed, L. Torkzadeh, A. Borumand Saeid and N. Mohtashamnia, Radical of filters in BL-algebras, Math. Log. Quart. 57, No. 2, (2011), 166-179 ].

This paper generalizes the idea of strongly associative hyperoperation introduced in [7] to the class of hyperrings. We introduce and investigate hyperrings of type 1, type 2 and SDIS. Moreover, we study some examples of these hyperrings and give a new kind of hyperrings called totally hyperrings. Totally hyperrings give us a characterization of Krasner hyperrings. Also, we investigate these strongly hyperoperations in hyperring of series.

K. Borsuk in 1979, at the Topological Conference in Moscow, introduced concept of the capacity of a compactum and asked some questions concerning properties of the capacity of compacta. In this paper, we give partial positive answers to three of these questions in some cases. In fact, by describing spaces homotopy dominated by Moore and Eilenberg-MacLane spaces, the capacities of a Moore space M(A,n) and an Eilenberg-MacLane space K(G,n) could be obtained. Also, we compute the capacity of wedge sum of finitely many Moore spaces of different degrees and the capacity of product of finitely many Eilenberg-MacLane spaces of different homotopy types. In particular, we compute the capacity of wedge sum of finitely many spheres of the same or different dimensions.

Let G be a finite non-abelian p-group and L(G) denotes the absolute center of G. Also, let \AutL(G) and \Autc(G) denote the group of all absolute central and the class preserving automorphisms of G, respectively. In this paper, we give a necessary and sufficient condition for G such that \Autc(G)=\AutL(G). We also characterize all finite non-abelian p-groups of order pn(n≤5), for which every absolute central automorphism is class preserving.

There are remarkable relations between the graded homological dimensions and the ordinary homological dimensions. In this paper, we study the injective dimension of a complex of graded modules and derive its some properties. In particular, we define the ∗dualizing complex for a graded ring and investigate its consequences.