Weakly Regular T-Seminearrings and Simple T-Seminearrings

Abstract

We define a Γ-seminearring as follows. Let (R,+) be a semigroup and Γ a nonempty set. Then R is called a Γ-seminearrng if there exists a mapping RxΓxR→R such that αα(bβc) = (ααb)βc and (α+b)αc = ααc for any α,b,c R and α, β Γ. Notice that Γ-seminearrings are generalization of Γ-nearrings, Γ-semirings and Γ-rings. Furthermore, Γ-seminearrings are generalization of seminearrings which are generalization of nearrings, semirings and rings. The first purpose of this thesis is to study general properties of Γ-seminearrings such as some results involving sub Γ-seminearrings, ideals of a Γ-seminearring and quotient Γ-seminearrings. As we know, regular rings, weakly regular semirings, simple nearrings and 0-simple Γ-rings have been studied. This leads to anotler purpose of this work which is to investigate regular Γ-seminearrings, weakly regular Γ-seminearrings, simple Γ-seminearrings and 0-simpleΓ-seminearrings.