On Jacobson Radical for Γ-Rings

Abstract

Jacobson radical of gamma rings is one of the most significant concept in the ring theory. In this paper we consider the Jacobson radical for gamma rings due to A.C. Paul and T.M. Abul Kalam Azad [5]. Some new characterizations are developed in this radical. The Jacobson Density Theorem and its converse Theorem are also proved here.

Some Generalizations of Semisimple Gamma Rings

“…Then the residual of in denoted by ( ) { }, which is a left ideal of [1]. An element of a ring is called idempotent if for some [3]. .…”

“…There are deferent of definitions of the regular ring. In [3] . Note that if a ring is regular in the sense of [10] and [3] ,then R is regular in the sense of [6].…”

Quasi-Injective Gamma Modules.

“…An element in ring is called left quasi-regular if there exists a in such that a a a a for each , an ideal of is left quasi-regular if each its elements is left quasi-regular [10]. …”

A Characterization of Jacobson Radical in Γ-Banach Algebras

“…Many interesting results on the internal properties of Jacobson radical for -rings were developed in [2][3][4][5] by different research workers. In [6], some of these results are extended to -algebras.…”

“…In [4], there is a characterization of Jacobson radical for -rings in terms of maximal regular left ideals.…”