An extensive generalized concept of classical ring set forth the notion of a gamma ring theory. As an emerging field of research, the research work of classical ring theory to the gamma ring theory has been drawn interest of many algebraists and prominent mathematicians over the world to determine many basic properties of gamma ring and to enrich the world of algebra. The different researchers ...

Let A be a semprime, right noetherian ring equipped with an automorphism α, and let B := A[[y;α]] denote the corresponding skew power series ring (which is also semiprime and right noetherian). We prove that the Goldie ranks of A and B are equal. We also record applications to induced ideals.

By means of a kind of new idea, we redefine some kinds of fuzzy ideals in a ring and investigate some of their related properties. The concepts of strong prime (semiprime) generalized fuzzy (bi-, quasi-) ideals in rings are introduced. In particular, we discuss the relationships between strong prime (resp., semiprime) generalized fuzzy (bi-, quasi-) ideals and prime (resp. semiprime) generalize...

Let / and g be functions of a set 5 into a semiprime ring R. A necessary and sufficient condition for / and g to satisfy f{s)xg{t) = g{s)xf{t) for all s, t e S, x e R is given. As an application, biderivations and commuting additive mappings of semiprime rings are characterized.

We prove in this note the following result. Let n > 1 be an integer and let R be an n!torsion-free semiprime ring with identity element. Suppose that there exists an additive mapping D : R→ R such that D(xn) =∑nj=1 xn− jD(x)x j−1is fulfilled for all x ∈ R. In this case, D is a derivation. This research is motivated by the work of Bridges and Bergen (1984). Throughout, R will represent an associ...

A von Neumann regular extension of a semiprime ring naturally deenes a epimorphic extension in the category of rings. These are studied, and four natural examples are considered, two in commutative ring theory, and two in rings of continuous functions.

Let I be a non-zero left ideal of r-ring M satisfying the condition alpha b beta c= c for all a,b,c, e and alpha, r. We show that contains non-trivial central if is semiprime which admits an appropriate derivations on I, also commutative prime admitting centralizing derivation I. next give some characterizations when generalized act as homomorphisms anti-homomorphisms or two-sided ideals gamma ...

Let R be a ring, σ an injective endomorphism of R and δ a σderivation of R. We prove that if R is semiprime left Goldie then the same holds for the Ore extension R[x;σ, δ] and both rings have the same left uniform dimension.

The purpose of this paper is to prove the following result: Let R be a 2torsion free semiprime ring and let T : R → R be an additive mapping, such that 2T (x) = T (x)x + xT (x) holds for all x ∈ R. In this case T is left and right centralizer.

In this paper, we introduce (α, α)-symmetric derivations and establish some interesting results and also extend an important result of J. Vukman by using (α, α)-derivation.