13,236 research outputs found
Zero-Divisors and Their Graph Languages
We introduce the use of formal languages in place of zero-divisor graphs used to study theoretic properties of commutative rings. We show that a regular language called a graph language can be constructed from the set of zero-divisors of a commutative ring. We then prove that graph languages are equivalent to their associated graphs. We go on to define several properties of graph languages
On annihilator graph of a finite commutative ring
‎The annihilator graph of a commutative ring is a simple undirected graph with the vertex set and two distinct vertices are adjacent if and only if ‎. ‎In this paper we give the sufficient condition for a graph to be complete‎. ‎We characterize rings for which is a regular graph‎, ‎we show that and we also characterize the rings for which has a cut vertex‎. ‎Finally we find the clique number of a finite reduced ring and characterize the rings for which is a planar graph‎
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