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.

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).

Let R be a commutative ring and M be an R-module. The essential GRAPH of M, denoted by EG(M) is a simple GRAPH with vertex set Z(M) \ Ann(M) and two distinct vertices x, y ∈ Z(M) \ Ann(M) are adjacent if and only if AnnM(xy) is an essential submodule of M. In this paper, we investigate the dominating set, the clique and the chromatic number and the metric dimension of the essential GRAPH for Noetherian MODULES. Let M be a Noetherian R-module such that |MinAssR(M)| = n ≥ 2 and let EG(M) be a connected GRAPH. We prove that EG(M) is a weakly prefect, that is, ω(EG(M)) = χ(EG(M)). Furthermore, it is shown that dim(EG(M)) = |Z(M)| − (| Ann(M)| + 2n), whenever r(Ann(M)) ̸= Ann(M) and dim(EG(M)) = |Z(M)| − (| Ann(M)| + 2n − 2), whenever r(Ann(M)) = Ann(M)

There are two approaches for simulating memory as well as learning in artificial intelligence; the functionalistic approach and the cognitive approach. The necessary condition to put the second approach into account is to provide a model of brain activity that contains a quite good congruence with observational facts such as mistakes and forgotten experiences. Given that human memory has a solid core that includes the components of our identity, our family and our hometown, the major and determinative events of our lives, and the countless repeated and accepted facts of our culture, the more we go to the peripheral spots the data becomes flimsier and more easily exposed to oblivion. It was essential to propose a model in which the topoGRAPHical differences are quite distinguishable. In our proposed model, we have translated this topoGRAPHical situation into quantities, which are attributed to the nodes. The result is an edge-weighted GRAPH with mass-based values on the nodes which demonstrates the importance of each atomic proposition, as a truth, for an intelligent being. Furthermore, it dynamically develops and modifies, and in successive phases, it changes the mass of the nodes and weight of the edges depending on gathered inputs from the environment.

For an R-module M and f ∈ M∗ = Hom(M, R), let Zf(M) and Regf(M) be the sets of all ZERO-DIVISORs elements and regular elements of M with re- spect to f, respectively. In this paper, we introduce the total GRAPH of M with respect to f, denoted by T(Γf(M)), which is the GRAPH with all the elements M as vertices, and for distinct elements m, n ∈ M, m and n are adjacent and only if m + n ∈ Zf(M). We also study the subGRAPHs Z(Γf(M)) and Reg(Γf(M)) with vertices Zf(M) and Regf(M), respectively.

Let R be a commutative ring and M be a unitary R- module. We define a semiprime submodule of a module and consider various properties of it. Also we define semi-radical of a submodule of a module and give a number of its properties. We define MODULES which satisfy the semi-radical formula (s.t .s.r.f) and present the existence of such a module.