Journal of Algebra and Related Topics
Volume:2 Issue: 2, Autumn 2014

The triple factorization of a group G has been studied recently showing that G=ABA for some proper subgroups A and B of G, the definition of rank-two geometry and rank-two coset geometry which is closely related to the triple factorization was defined and calculated for abelian groups. In this paper we study two infinite classes of non-abelian finite groups D2n and PSL(2,2n) for their triple factorizations by finding certain suitable maximal subgroups, which these subgroups are define with original generators of these groups. The related rank-two coset geometries motivate us to define the rank-two coset geometry graphs which could be of intrinsic tool on the study of triple factorization of non-abelian groups.

Let R be a commutative ring. In this paper, by using algebraic properties of R, we study the Hase digraph of prime ideals of R.

For an n-gon with vertices at points 1,2,⋯,n, the Betti numbers of its suspension, the simplicial complex that involves two more vertices n and n, is known. In this paper, with a constructive and simple proof, we generalize this result to find the minimal free resolution and Betti numbers of the S-module S/I where S=K[x1,⋯,xn] and I is the associated ideal to the generalized suspension of it in the Stanley-Reisner sense. Applications to Stanley-Reisner ideals and simplicial complexes are considered.

Primary-like and weakly primary-like submodules are two new generalizations of primary ideals from rings to modules. In fact, the class of primary-like submodules of a module lie between primary submodules and weakly primary-like submodules properly. In this note, we show that these three classes coincide when their elements are submodules of a multiplication module and satisfy the primeful property.

In this paper we introduce a new definition of the first non-abelian cohomology of topological groups. We relate the cohomology of a normal subgroup N of a topological group G and the quotient G/N to the cohomology of G. We get the inflation-restriction exact sequence. Also, we obtain a seven-term exact cohomology sequence up to dimension 2. We give an interpretation of the first non-abelian cohomology of a topological group by the notion of a principle homogeneous space.

In this paper we introduce the concept of weakly prime ternary subsemimodules of a ternary semimodule over a ternary semiring and obtain some characterizations of weakly prime ternary subsemimodules. We prove that if N is a weakly prime subtractive ternary subsemimodule of a ternary R-semimodule M, then either N is a prime ternary subsemimodule or (N:M)(N:M)N=0. If N is a Q-ternary subsemimodule of a ternary R-semimodule M, then a relation between weakly prime ternary subsemimodules of M containing N and weakly prime ternary subsemimodules of the quotient ternary R-semimodule M/N(Q) is obtained.