A conception of zero-divisor graph for categories of modules

作者：Amini Afshin^*; Amini Babak; Momtahan Ehsan

来源：Journal of Algebra and Its Applications, 2016, 15(1): 1650012.

DOI：10.1142/S0219498816500122

• Category of modules
• perfect ring
• diameter of a graph
• complements of a vertex

We introduce and study zero-divisor graphs in categories of left modules over a ring R, i.e. R-MOD. The vertices of Gamma(R-MOD) consist of all nonzero morphisms in R-MOD which are not isomorphisms. Two vertices f and g are adjacent if f circle g = 0 or g circle f = 0. We observe that these graphs are connected and their diameter is equal or less than four. We prove that diam Gamma(R-MOD) = 3 if and only if R is a right and left perfect ring and R/J(R) is simple artinian. We also characterize all vertices with complements and that when a kernel or a co-kernel can be a complement for a morphism. Some discussions will be made on radius of these graphs, their clique and chromatic numbers.

• 出版日期2016-2