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.

‎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.

Let $R$ be a commutative ring with identity, $M$ be a unital $R$-module and let $L$ be a complete Heyting algebra. In this paper, among results on colon structures of $L$-neutrosophic submodules and $L$-neutrosophic ideals, we introduce and study the notion of primary (and prime) $L$-neutrosophic submodules and give connections with primary (prime) behavior of its $t$, $i$ and $f$ components. Then, for a multiplicatively closed subset $S$ of $R$, we define the notion of localization formation for an $L$-neutrosophic submodule $\lambda$ of $M$ and study its behavior. Some types of $L$-neutrosophic quotients will also be investigated.

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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.