The annihilating-submodule graph of modules over commutative rings

Let M be a module over a commutative ring R. In this paper, we continue our study of annihilating-submodule graph AG(M) which was introduced in (The Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42 (2014), 3283{3296). AG(M) is a (undirected) graph in which a nonzero submodule N of M is a vertex if and only if there exists a nonzero proper submodule K of M such that NK = (0), where NK, the product of N and K, is defined by (N : M)(K : M)M and two distinct vertices N and K are adjacent if and only if NK = (0). We obtain useful characterizations for those modules M for which either AG(M) is a complete (or star) graph or every vertex of AG(M) is a prime (or maximal) submodule of M. Moreover, we study coloring of annihilating-submodule graphs.Comment: 14 pages, 0 figures. arXiv admin note: text overlap with arXiv:0808.3189 by other author

Determining Corresponding Artinian Rings to Zero-Divisor Graphs

We introduce Anderson\u27s and Livingston\u27s definition of a zero-divisor graph of a commutative ring. We then redefine their definition to include looped vertices, enabling us to visualize nilpotent elements. With this new definition, we examine the algebraic and graph theoretic properties of different types of Artinian rings, culminating in an algorithm that determines the corresponding Artinian rings to a zero-divisor graph. We also will explore and develop an algorithm for a specific case of Artinian rings, and we will conclude by examining the uniqueness of zero-divisor graphs