Abstract
Let L be a multiplicative lattice and M be a lattice module over L. In this paper, we assign a graph to M called residual division graph RG(M) in which the element N ∈ M is a vertex if there exists 0M ≠ P ∈ M such that NP = 0M and two vertices N1, N2 are adjacent if N1N2 = 0M (where N1N2 = (N1 : IM)(N2 : IM)IM). It is proved that such a graph with the greatest element IM which does not belong to the vertex set is nonempty if and only if M is a prime lattice module. Also, we provide conditions such that RG(M) is isomorphic to a subgraph of Zariski topology graph ĢX(M) with respect to X.
Highlights
In our everyday life, we found that numerous issues are dealt with the assistance of graphs
Beck first presented the zero-divisor graphs of a commutative ring with unity. is examination of the coloring of a commutative ring was further studied by Anderson and Naseer
Behboodi et al studied the annihilating-ideal graph AG(R) with vertex set V(AG(R)) contain all those ideals of ring R whose annihilators are nonzero(see [3, 4]). ereafter, Ansari-Toroghy et al expanded this work for R-module M, where R is a commutative ring. ey studied the algebraic as well as topological properties of M with the help of annihilating-submodule graph and the Zariski topology graph
Summary
We found that numerous issues are dealt with the assistance of graphs. E ring structure is firmly associated with ideals more than elements and so it has the right to present a graph with vertices as ideals instead of elements. In the recent paper [5], Ansari-Toroghy and Habibi have highlighted the closed sets in Zariski topology on prime spectrum Spec (M) of R-module M and defined new graph called Zariski topology graph G(τT), where T ⊆ Spec(M) which is nonempty. Roughout the paper, M denotes a lattice module over a multiplicative lattice L, and for N, K ∈ M, we define.
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