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THE ZERO-DIVISOR GRAPH UNDER A GROUP ACTION IN A COMMUTATIVE RING
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 Title & Authors
THE ZERO-DIVISOR GRAPH UNDER A GROUP ACTION IN A COMMUTATIVE RING
Han, Jun-Cheol;
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 Abstract
Let R be a commutative ring with identity, X the set of all nonzero, nonunits of R and G the group of all units of R. We will investigate some ring theoretic properties of R by considering (R), the zero-divisor graph of R, under the regular action on X by G as follows: (1) If R is a ring such that X is a union of a finite number of orbits under the regular action on X by G, then there is a vertex of (R) which is adjacent to every other vertex in (R) if and only if R is a local ring or where F is a field; (2) If R is a local ring such that X is a union of n distinct orbits under the regular action of G on X, then all ideals of R consist of {{0}, J, , , , R} where J is the Jacobson radical of R; (3) If R is a ring such that X is a union of a finite number of orbits under the regular action on X by G, then the number of all ideals is finite and is greater than equal to the number of orbits.
 Keywords
zero-divisor graph;regular action;orbit;local ring;
 Language
English
 Cited by
 References
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