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 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.

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.

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.

Let $R$ and $S$ be commutative rings and $M$ an $(R,S)$-module. A proper $(R,S)$-SUBMODULE $P$ of $M$ is called left WEAKLY jointly prime if for each $(R,S)$-SUBMODULE $N$ of $M$ and elements $a,b$ of $R$ such that $abNS\subseteq P$ implies either $aNS\subseteq P$ or $bNS\subseteq P$. This paper defines left WEAKLY jointly prime $(R,S)$-modules and presents some of their properties. On the other hand, a ring $R$ is called fully prime if each proper ideal of $R$ is prime. We extend this fact to $(R,S)$-modules. An $(R,S)$-module $M$ is called fully left WEAKLY jointly prime if each proper $(R,S)$-SUBMODULE of $M$ is left WEAKLY jointly prime. Moreover, we present some properties of fully left WEAKLY jointly prime $(R,S)$-modules. At the end of this paper, we present our main results about the necessary and sufficient conditions for an arbitrary $(R,S)$-module to be fully left WEAKLY jointly prime.