Let R be a commutative ring and M an R -module. In this article, we introduce a new gen-eralization of the ANNIHILATING-ideal GRAPH of commutative rings to modules. The ANNIHILATING sub module GRAPH of M, denoted by G (M), is an undirected GRAPH with vertex set A * (M) and two distinct elements Nand K of A * (M) are adjacent if N * K=0. In this paper we show that G (M) is a connected GRAPH, diam (G (M)) £ 3, and gr (G (M)) £ 4 if G (M) contains a cycle. Moreover, G (M) is an empty GRAPH if and only if ann (M) is a prime ideal of R and A * (M) ¹ S (M) / {0} if and only if M is a uniform R-module, ann (M) is a semi-prime ideal of R and A * (M) ¹ S (M) / {0}. Furthermore, R is a eld if and only if G (M) is a complete GRAPH, for every M Î R - Mod. If R is a domain, for every divisible module M Î R-Mod, G (M) is a complete GRAPH with A * (M) =S (M) / {0}. Among other things, the properties of a reduced R -module M are investigated when G (M) is a bipartite GRAPH.

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Consider a commutative ring $R$ with a non-zero identity $1\neq 0$, and let $M$ be a non-zero unitary module over $R$. In this document, our goal is to present the sum-ANNIHILATING essential SUBMODULE GRAPH $\mathbb{AE}^{0}_{R}(M)$ and its subGRAPH $\mathbb{AE}^{1}_{R}(M)$ of a module $M$ over a commutative ring $R$ which is described in the following way: The vertex set of GRAPH $\mathbb{AE}^{0}_{R}(M)$ (resp., $\mathbb{AE}^{1}_{R}(M)$) is the collection of all (resp., non-zero proper) ANNIHILATING SUBMODULEs of $M$ and two separate ANNIHILATING SUBMODULEs $N$ and $K$ are connected anytime $N+K$ is essential in $M$. We study and investigate the basic properties of GRAPHs $\mathbb{AE}^{i}_{R}(M)$ ($i=0, 1$) and will present some related results. Additionally, we explore how the properties of GRAPHs interact with the algebraic structures they represent.

Let M be a module over a commutative ring R. We continue our study of strongly ANNIHILATING SUBMODULE GRAPH SAG(M) introduced in [11]. In addition to providing the more properties of this GRAPH, we introduce the subGRAPH SAG∗ (M) of SAG(M) and compare the properties of SAG∗ (M) with SAG(M) and AG(M) (the ANNIHILATING SUBMODULE GRAPH of M introduced in [4]).

In this article, we give several generalizations of the concept of ANNIHILATING an ideal GRAPH over a commutative ring with identity to modules. We observe that, over a commutative ring, R, AG (RM) is connected, and diamAG (RM) ≤ 3. More-over, if AG (RM) contains a cycle, then grAG (RM) ≤ 4. Also for an R-module M with A (M) ̸ = S(M) \ {0}, A (M) = ∅ , if and only if M is a uniform module, and ann(M) is a prime ideal of R.

Let A be a commutative ring with unity. The ANNIHILATING GRAPH of A, denoted by Gð AÞ , is a GRAPH whose vertices are all non-trivial ideals of A and two distinct vertices I and J are adjacent if and only if Annð IÞ Annð JÞ ¼ 0. For every commutative ring A, we study the diameter and the girth of Gð AÞ . Also, we prove that if Gð AÞ is a triangle-free GRAPH, then Gð AÞ is a bipartite GRAPH. Among other results, we show that if Gð AÞ is a tree, then Gð AÞ is a star or a double star GRAPH. Moreover, we prove that the ANNIHILATING GRAPH of a commutative ring cannot be a cycle. Let n be a positive integer number. We classify all integer numbers n for which Gð ZnÞ is a complete or a planar GRAPH. Finally, we compute the domination number of Gð ZnÞ .

Let R be a commutative ring with identity and M be an R-module. The zero divisor GRAPH of M is denoted by 􀀀 (M). In this study, we are going to generalize the zero divisor GRAPH 􀀀 (M) to SUBMODULE-based zero divisor GRAPH 􀀀 (M, N) by replacing elements whose product is zero with elements whose product is in some SUBMODULE N of M. The main objective of this paper is to study the interplay of the properties of SUBMODULE N and the properties of 􀀀 (M, N).

The ANNIHILATING-ideal GRAPH of a commutative ring R with unity is de-, ned as the GRAPH AG(R) whose vertex set is the set of all non-zero ideals with non-zero annihilators and two distinct vertices I and J are adjacent if and only if IJ = 0. Nikan-dish et. al. proved that AG(Zn) is weakly perfect. In this short paper, we characterize n for which AG(Zn) is perfect.

Let R be a commutative ring and A(R) be the set of all ideals with non-zero annihilators. Assume that A (R) = A(R)⧹ f(0)g and F(R) denote the set of all  nitely generated ideals of R. In this paper, we introduce and investigate the  nitely generated ANNIHILATING-ideal GRAPH of R, denoted by AGF (R). It is the (undirected) GRAPH with vertices AF (R) = A (R) \ F(R) and two distinct vertices I and J are adjacent if and only if IJ = (0). First, we study some basic properties of AGF (R). For instance, it is shown that if R is not a domain, then AGF (R) has ascending chain condition on vertices if and only if R is Noetherian. We characterize all rings for which AGF (R) is a  nite, complete, star or bipartite GRAPH. Next, we study diameter and girth of AGF (R). It is proved that diam(AGF (R)) ⩽ diam(AG(R)) and gr(AGF (R)) = gr(AG(R)):