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THE ZERO-DIVISOR GRAPH UNDER GROUP ACTIONS IN A NONCOMMUTATIVE RING
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 Title & Authors
THE ZERO-DIVISOR GRAPH UNDER GROUP ACTIONS IN A NONCOMMUTATIVE RING
Han, Jun-Cheol;
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 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 of a noncommutative ring R as follows: (1) if has no sources and no sinks, then is connected and diameter of , denoted by diam() (resp. girth of , denoted by g()) 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 is connected and diam() (resp. g()) is equal to or less than 3, in addition, if R is local, then there is a vertex of which is adjacent to every other vertices in ; (3) if R is unit-regular, then is connected and diam() (resp. g()) is equal to or less than 3. Next, we investigate the graph automorphisms group of where is the ring of 2 by 2 matrices over the galois field (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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