科研之友
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摘要
In this paper, a kind of graph structure Gamma(N)(R) of a ring R is introduced, and the interplay between the ring-theoretic properties of R and the graph-theoretic properties of Gamma(N)(R) is investigated. It is shown that if R is Artinian or commutative, then Gamma(N)(R) is connected, the diameter of Gamma(N)(R) is at most 3; and if Gamma(N)(R) contains a cycle, then the girth of Gamma(N)(R) is riot more than 4; moreover, if R is non-reduced, then the girth of Gamma(N)(R) is 3. For a finite commutative ring R, it is proved that the edge chromatic number of Gamma(N)(R) is equal to the maximum degree of Gamma(N)(R) unless R is a nilpotent ring with even order. It is also shown that, with two exceptions, if R is a finite reduced commutative ring and S is a commutative ring which is not an integral domain and Gamma(N)(R) similar or equal to Gamma(N)(S), then R similar or equal to S. If R and S are finite non-reduced commutative rings and Gamma(N)(R) similar or equal to Gamma(N)(S), then vertical bar R vertical bar = vertical bar S vertical bar and vertical bar N(R)vertical bar = vertical bar N(S)vertical bar.
