Journal of Algebraic Structures and Their Applications
Volume:2 Issue: 2, Summer - Autumn 2015

In this paper we give a characterization for all semigroups whose square is a group. Moreover, we axiomatize such semigroups and study some relations between the class of these semigroups and Grouplikes, introduced by the author. Also, we observe that this paper characterizes and axiomatizes a class of Homogroups (semigroups containing an ideal subgroup). Finally, several equivalent conditions for a semigroup S with S2≤S (the square-group property) will be considered.

The rings considered in this article are commutative with identity which admit at least two nonzero annihilating ideals. Let R be a ring. Let A(R) denote the set of all annihilating ideals of R and let A(R)∗=A(R)∖{(0)}. The annihilating-ideal graph of R, denoted by AG(R) is an undirected simple graph whose vertex set is A(R)∗ and distinct vertices I,J are joined by an edge in this graph if and only if IJ=(0). The aim of this article is to classify rings R such that (AG(R))c ( that is, the complement of AG(R)) is connected and admits a cut vertex.

In this paper, we define the notions of ultra and involution ideals in BCK-algebras. Then we get the relation among them and other ideals as (positive) implicative, associative, commutative and prime ideals. Specially, we show that in a bounded implicative BCK-algebra, any involution ideal is a positive implicative ideal and in a bounded positive implicative lower BCK-semilattice, the notions of prime ideals and ultra ideals are coincide.

Assume that (N,L), is a pair of finite dimensional nilpotent Lie algebras, in which L is non-abelian and N is an ideal in L and also M(N,L) is the Schur multiplier of the pair (N,L). Motivated by characterization of the pairs (N,L) of finite dimensional nilpotent Lie algebras by their Schur multipliers (Arabyani, et al. 2014) we prove some properties of a pair of nilpotent Lie algebras and generalize results for a pair of non-abelian nilpotent Lie algebras.

In this paper we characterize all 2×2 idempotent and nilpotent matrices over an integral domain and then we characterize all 2×2 strongly nil-clean matrices over a PID. Also, we determine when a 2×2 matrix over a UFD is nil-clean.

In this article we introduce the concept of z∘-filter on a topological space X. We study and investigate the behavior of z∘-filters and compare them with corresponding ideals, namely, z∘-ideals of C(X), the ring of real-valued continuous functions on a completely regular Hausdorff space X. It is observed that X is a compact space if and only if every z∘-filter is ci-fixed. Finally, by using z∘-ultrafilters, we prove that any arbitrary product of i-compact spaces is i-compact.