Some notes on first strongly graded rings

Semisimple Strongly Graded Rings

Let G be a finite group and R a strongly G-graded ring. The question of when R is semisimple (meaning in this paper semisimple artinian) has been studied by several authors. The most classical result is Maschke’s Theorem for group rings. For crossed products over fields there is a satisfactory answer given by Aljadeff and Robinson [3]. Another partial answer for skew group rings was given by Al...

Some notes on CN rings

The main results: A ring R is CN if and only if for any x ∈ N(R) and y ∈ R, ((1+x)y)n+k = (1+x)n+kyn+k, where n is a fixed positive integer and k = 0, 1, 2; (2) Let R be a CN ring and n ≥ 1. If for any x, y ∈ R\N(R), (xy)n+k = xn+kyn+k, where k = 0, 1, 2, then R is commutative; (3) Let R be a ring and n ≥ 1. If for any x ∈ R\N(R) and y ∈ R, (xy)k = xkyk, k = n, n + 1, n + 2, then R is commutati...

Some Notes on Semiabelian Rings

Some classes of strongly clean rings

A ring $R$ is a strongly clean ring if every element in $R$ is the sum of an idempotent and a unit that commutate. We construct some classes of strongly clean rings which have stable range one. It is shown that such cleanness of $2 imes 2$ matrices over commutative local rings is completely determined in terms of solvability of quadratic equations.

On the Jacobson radical of strongly group graded rings

For any non-torsion group G with identity e, we construct a strongly G-graded ring R such that the Jacobson radical J(Re) is locally nilpotent, but J(R) is not locally nilpotent. This answers a question posed by Puczy lowski.