‎In this paper‎, ‎our aim is to introduce and study the essential submodules of an $R$-module $M$ relative to an arbitrary submodule $T$ of $M$‎. ‎Let $T$ be an arbitrary submodule of an $R$-module $M$‎, ‎then we say that a submodule $N$ of $M$ is an essential submodule of $M$ relative to $T$‎, ‎whenever for every submodule $X$ of $M$‎, ‎$N\cap X\subseteq T$ implies that‎ ‎$(T:M)\subseteq ^{e}{\rm Ann}(X)$‎. ‎We investigate some new results concerning to this class of submodules‎. ‎Among various results we prove that for a faithful multiplication $R$-module $M$‎, ‎if the submodule $N$ of $M$ is an essential submodule of $M$ relative to $T$‎, ‎then $(N:M)$ is an essential ideal of $R$ relative to $(T:M)$‎. ‎The converse is true if $M$ is moreover a finitely generated module‎.

The concept of essential submodules is a well known concept. In this paper we try to replace an arbitrary submodule of M, say T, instead of 0 in the definition of essential submodules. By this, essential submodules are precisely {0}-essential submodules. For a submodule K of right R-module M, we have Íess M if and only if (K: m) is annr (m)-essential right ideal of R, for each mÎM \ {0}. Among other things, this generalization of essential submodules gives a necessary and sufficient condition for M/T being finitely co-generated.

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 R be a commutative ring with identity and M be a unitaryR-module. A proper submodule N of M is 2- absorbing if r1, r2, r3 Î R, m Î M with r1r2r3m Î M implies r1r2m Î N or r1r3m Î N or r2r3m Î N. Let j: S (M) ® S (M) È {f} be a function where S (M) is the set of all submodules of M. We call a proper submodule No f M a j-2-absorbing submodule if r1, r2, r3 Î R, m Î M with r1r2r3m Î N -j (N) implies that r1r2m Î N or r1r3m Î N or r2r3m Î N. We want to extend 2-absorbing ideals to j-2-absorbing submodules and we show that j-2-absorbing submodules enjoy analogs of many of the properties of 2-absorbing ideals.

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 domain with quotiont field K, and let N be a submodule of an R -module M. We say that N is powerful (strongly primary) if x, yÎK and xyMÍN, then xÎR or yÎR (xMÍN or ynMÍN for some n³1). We show that a submodule with either of these properties is comparable to every prime submodule of M, also we show that an R -module M admits a powerful submodule if and only if it admits a strongly primary submodule. Finally we study finitely generated torsion free modules over domain each of whose prime submodules are strongly primary.

In this article, we extend the concept of divisors to ideals of Noetherian rings, more generally, to submodules of finitely generated modules over Noetherian rings. For a submodule $N$ of a finitely generated module $M$ over a Noetherian ring, we say a submodule $K$ of $M$ is a regular divisor of $N$ in $M$ if $K$ occurs in a regular prime extension filtration of $M$ over $N$. We show that a submodule $N$ of $M$ has only a finite number of regular divisors in $M$. We also show that an ideal $\mathfrak b$ is a regular divisor of a non-zero ideal $\mathfrak a$ in a Dedekind domain $R$ if and only if $\mathfrak b$ contains $\mathfrak a$. We characterize regular divisors using some ordered sequences of prime ideals and study their various properties. Lastly, we formulate a method to compute the number of regular divisors of a submodule by solving a combinatorics problem.

The submodules with the property of the title (a submodule N of an R -module M is called strongly dense in M, denoted by N£sd M, if for any index set I, ΠIN£d ΠIM) are introduced and fully investigated. It is shown that for each submoduleN of M there exists the smallest subsetD′ ÍM such that N+D′ is a strongly dense submodule of M and D′ ∩ N=0. We also introduce a class of modules in which the two concepts of strong essentiality and strong density coincide. It is also shown that for any module M, dense submodules in M are strongly dense if and only if M £sd ~E (M), where ~E (M) is the rational hull of M. It is proved that R has no strongly dense left ideal if and only if no nonzeroelement of every cyclic R -module M has a strongly dense annihilator in R. Finally, some properties and new concepts related to strong density are studied.