Journal of Algebra and Related Topics
Volume:5 Issue: 1, Summer 2017

In this paper first we introduce the notion of weak Γ -(semi)hypergroups, next some classes of equivalence relations which are called good regular and strongly good regular relations are defined. Then we investigate some properties of this kind of relations on weak Γ -(semi)hypergroups.

Let Q be an inverse semigroup. A subsemigroup S of Q is a left I-order in Q and Q is a semigroup of left I-quotients of S if every element q∈Q can be written as q=a −1 b for some a,b∈S . If we insist on a and b being \er -related in Q , then we say that S is straight in Q . We characterize semigroups which are left I-quotients of left regular bands of right cancellative monoids with certain conditions.

The rings considered in this article are commutative rings with identity 1≠0 . The aim of this article is to define and study the exact annihilating-ideal graph of commutative rings. We discuss the interplay between the ring-theoretic properties of a ring and graph-theoretic properties of exact annihilating-ideal graph of the ring.

In this paper, we introduce a new algebraic structure, called a UP-algebra (UP means the University of Phayao) and a concept of UP-ideals, UP-subalgebras, congruences and UP-homomorphisms in UP-algebras, and investigated some related properties of them.
We also describe connections between UP-ideals, UP-subalgebras, congruences and UP-homomorphisms, and show that the notion of UP-algebras is a generalization of KU-algebras.

This paper deals with some results concerning the notion of extended ideal based zero divisor graph Γ ¯ ¯ ¯ I (R) for an ideal I of a commutative ring R and characterize its bipartite graph. Also, we study the properties of an annihilator of Γ¯ Γ¯I(R)

The rings considered in this article are commutative with identity. This article is motivated by the work on comaximal graphs of rings. In this article, with any ring R , we associate an undirected graph denoted by G(R) , whose vertex set is the set of all elements of R and distinct vertices x,y are joined by an edge in G(R) if and only if Rx∩Ry=Rxy . In Section 2 of this article, we classify rings R such that G(R) is complete and we also consider the problem of determining rings R such that χ(G(R))=ω(G(R))