A ZERO-DIVISOR GRAPH FOR MODULES WITH RESPECT TO THEIR (FIRST) DUAL

作者：Baziar M^*; Momtahan E; Safaeeyan S

来源：Journal of Algebra and Its Applications, 2013, 12(2): 1250151.

DOI：10.1142/S0219498812501514

• Zero-divisor graph
• clique number
• chromatic number
• module

Let M be an R-module. We associate an undirected graph Gamma(M) to M in which nonzero elements x and y of M are adjacent provided that xf(y) = 0 or yg(x) = 0 for some nonzero R-homomorphisms f, g is an element of Hom(M, R). We observe that over a commutative ring R, Gamma(M) is connected and diam(Gamma(M)) <= 3. Moreover, if Gamma(M) contains a cycle, then gr(Gamma(M)) <= 4. Furthermore if vertical bar Gamma(M)vertical bar >= 1, then Gamma(M) is finite if and only if M is finite. Also if Gamma(M) = empty set, then any nonzero f is an element of Hom(M, R) is monic (the converse is true if R is a domain). For a nonfinitely generated projective module P we observe that Gamma(P) is a complete graph. We prove that for a domain R the chromatic number and the clique number of Gamma(M) are equal. When R is self-injective, we will also observe that the above adjacency defines a covariant functor between a subcategory of R-MOD and the Category of graphs.

• 出版日期2013-3