THE ZERO-DIVISOR GRAPH UNDER GROUP ACTIONS IN A NONCOMMUTATIVE RING

Title & Authors
THE ZERO-DIVISOR GRAPH UNDER GROUP ACTIONS IN A NONCOMMUTATIVE RING
Han, Jun-Cheol;

Abstract
Let R be a ring with identity, X the set of all nonzero, nonunits of R and G the group of all units of R. First, we investigate some connected conditions of the zero-divisor graph $\small{\Gamma(R)}$ of a noncommutative ring R as follows: (1) if $\small{\Gamma(R)}$ has no sources and no sinks, then $\small{\Gamma(R)}$ is connected and diameter of $\small{\Gamma(R)}$, denoted by diam($\small{\Gamma(R)}$) (resp. girth of $\small{\Gamma(R)}$, denoted by g($\small{\Gamma(R)}$)) is equal to or less than 3; (2) if X is a union of finite number of orbits under the left (resp. right) regular action on X by G, then $\small{\Gamma(R)}$ is connected and diam($\small{\Gamma(R)}$) (resp. g($\small{\Gamma(R)}$)) is equal to or less than 3, in addition, if R is local, then there is a vertex of $\small{\Gamma(R)}$ which is adjacent to every other vertices in $\small{\Gamma(R)}$; (3) if R is unit-regular, then $\small{\Gamma(R)}$ is connected and diam($\small{\Gamma(R)}$) (resp. g($\small{\Gamma(R)}$)) is equal to or less than 3. Next, we investigate the graph automorphisms group of $\small{\Gamma(Mat_2(\mathbb{Z}_p))}$ where $\small{Mat_2(\mathbb{Z}_p)}$ is the ring of 2 by 2 matrices over the galois field $\small{\mathbb{Z}_p}$ (p is any prime).
Keywords
connected (resp. complete) zero-divisor graph;left (resp. right) regular action;orbit;graph automorphisms group;
Language
English
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