On r-Noncommuting Graph of Finite Rings

Let R be a finite ring and r∈R. The r-noncommuting graph of R, denoted by ΓRr, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if and only if [x,y]≠r and [x,y]≠−r. In this paper, we obtain expressions for vertex degrees and show that ΓRr is neither a regular graph nor a lollipop graph if R is noncommutative. We characterize finite noncommutative rings such that ΓRr is a tree, in particular a star graph. It is also shown that ΓR1r and ΓR2ψ(r) are isomorphic if R1 and R2 are two isoclinic rings with isoclinism (ϕ,ψ). Further, we consider the induced subgraph ΔRr of ΓRr (induced by the non-central elements of R) and obtain results on clique number and diameter of ΔRr along with certain characterizations of finite noncommutative rings such that ΔRr is n-regular for some positive integer n. As applications of our results, we characterize certain finite noncommutative rings such that their noncommuting graphs are n-regular for n≤6.

Uniformly Primal Submodule over Noncommutative Ring

Let R be an associative ring with identity and M be a unitary right R-module. A submodule N of M is called a uniformly primal submodule provided that the subset B of R is uniformly not right prime to N , if there exists an element s ∈ M − N with sRB ⊆ N .The set adj N = r ∈ R | mRr ⊆ N for some m ∈ M is uniformly not prime to N .This paper is concerned with the properties of uniformly primal submodules. Also, we generalize the prime avoidance theorem for modules over noncommutative rings to the uniformly primal avoidance theorem for modules.

On the diameter of compressed zero-divisor graphs of ore extensions

This paper continues the ongoing effort to study the compressed zero-divisor graph over noncommutative rings. The purpose of our paper is to study the diameter of the compressed zero-divisor graph of Ore extensions and give a complete characterization of the possible diameters of ?E(R[x; ?,?]), where the base ring R is reversible and also have the (?,?)-compatible property. Also, we give a complete characterization of the diameter of ?E (R[[x;?]]), where R is a reversible, ?-compatible and right Noetherian ring. By some examples, we show that all of the assumptions ?reversiblity?, ?(?,?)-compatiblity? and ?Noetherian? in our main results are crucial.