Let R be a commutative ring with unity and M be a unitary R module. Let Nil(M) be the set of all nilpotent elements of M. The entire nilpotent element graph of M over R is an undirected graph E(G(M)) with vertex set as M and any two distinct vertices x and y are adjacent if and only if x + y ∈ Nil(M). In this paper we attempt to study the domination in the graph E(G(M)) and investigate the domination number as well as bondage number of E(G(M)) and its induced subgraphs N(G(M)) and Non(G(M)). Some domination parameters of E(G(M)) are also studied. It has been showed that E(G(M)) is excellent, domatically full and well covered under certain conditions.

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Let M be a module over a commutative ring R. In this paper we generalize some annihilator conditions on rings to modules. Denote by N il.M / the set of all nilpotent elements of M. M is said to be weak Armendariz if f .x/m.x/ D 0, where f .x/ D P n i D0 a i x i 2 ROExnf0g and m.x/ D P k j D0 m j x j 2 M OExnf0g, then a i m j 2 N il.M / for each i D 0; 1; :::; n and j D 0; 1; :::; k. We prove that the class of these modules are closed under direct sum, finite product and localization. We also prove that if M N is weak Armendariz, then so is M. Furthermore, we show that if D-module M is torsion, for a domain D, then M is weak Armendariz if and only if T .M / is weak Armendariz, where T .M / is the torsion submodule of M .

In this paper, we introduce the entire graph of a module with respect to a submodule, as a generalization of the total graph of a commutative ring in the sense of Anderson–Badawi. Let [Formula: see text] be an [Formula: see text]-module over a commutative ring [Formula: see text], [Formula: see text] a submodule of [Formula: see text] and [Formula: see text] the set of all elements [Formula: see text] such that [Formula: see text]. The entire graph of [Formula: see text] with respect to [Formula: see text] denoted by [Formula: see text] whose vertices are all elements of [Formula: see text] and two distinct elements [Formula: see text] are adjacent if and only if [Formula: see text]. Also, we study two (induced) subgraphs [Formula: see text] and [Formula: see text] of [Formula: see text], with vertices [Formula: see text] and [Formula: see text], respectively.