Residual Division Graph of Lattice Modules

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 0 M ≠ P ∈ M such that N P = 0 M and two vertices N 1 , N 2 are adjacent if N 1 N 2 = 0 M (where N 1 N 2 = ( N 1 : I M )( N 2 : I M ) I M ). It is proved that such a graph with the greatest element I M 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 R G ( M ) is isomorphic to a subgraph of Zariski topology graph Ģ X ( M ) with respect to X .

Abstract licence: CC BY

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