Clean graph of a ring

Let [Formula: see text] be a ring (not necessarily commutative) with identity. The clean graph [Formula: see text] of a ring [Formula: see text] is a graph with vertices in form [Formula: see text], where [Formula: see text] is an idempotent and [Formula: see text] is a unit of [Formula: see text]; and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. In this paper, we focus on [Formula: see text], the subgraph of [Formula: see text] induced by the set [Formula: see text] is a nonzero idempotent element of [Formula: see text]. It is observed that [Formula: see text] has a crucial role in [Formula: see text]. The clique number, the chromatic number, the independence number and the domination number of the clean graph for some classes of rings are determined. Moreover, the connectedness and the diameter of [Formula: see text] are studied.

I-Reversible rings

We study rings in which [Formula: see text] nonzero idempotent implies [Formula: see text] is also an idempotent. We call such rings i-reversible. Besides studying the basic properties of i-reversible rings, we characterize i-reversible triangular matrix rings, i-reversible matrix rings over commutative rings and i-reversible exchange rings.

Simple semigroup graded rings

We show that if R is a, not necessarily unital, ring graded by a semigroup G equipped with an idempotent e such that G is cancellative at e, the nonzero elements of eGe form a hypercentral group and Re has a nonzero idempotent f, then R is simple if and only if it is graded simple and the center of the corner subring f ReGe f is a field. This is a generalization of a result of Jespers' on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity of a, not necessarily unital, partial skew group ring by a hypercentral group. Thereby, we generalize a very recent result of Gonçalves'. We also point out how Jespers' result immediately implies a generalization of a simplicity result, recently obtained by Baraviera, Cortes and Soares, for crossed products by twisted partial actions.

The Nilpotent filtration and the action of automorphisms on the cohomology of finite p-groups

We study H*(P), the mod p cohomology of a finite p-group P, viewed as an $\F_p[Out(P)]$–module. In particular, we study the conjecture, first considered by Martino and Priddy, that, if e ∈ $\F_p[Out(P)]$ is a nonzero idempotent, then the Krull dimension of eH*(P) equals the rank of P. We prove this for all p-groups when p is odd, and for many 2–groups.

A CONDITION FOR SIMPLE RING IMPLYING FIELD II

It is shown that if $R$ is a simple ring with identity 1 and with a nonzero idempotent $e$ and satisfies the condition $(P_2)_e$ : $(P_2)_e$ If $e- (a_1b_1+a_2b_2)$ is a right (left)zero divisor in $R$, then so is $e- (b_1a_1+b_2a_2)$. then $R$ is a field.Thus if $R$ is a simple ring then $eRe$ is a field for every nonzero idempotent $e$ in $R$ if it exists and $eRe$ satisfies $(P_2)_e$. We also discuss the above property for the simple ring case by eliminating the identity 1.