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 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.

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.

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.

An $R$-module $M$ is called torsion-free, if $rx=0$ for $r\in R$ and $x\in M$ implies that $r=0$ or $x=0$. In this paper, we introduce the notions semi torsion-free modules and quasi torsion-free modules. We show that a SUBMODULE $N$ of an $R$-module $M$ is a $P$-PRIMARY SUBMODULE if and only if $\dfrac{R}{P}$-module $\dfrac{M}{N}$ is semi torsion-free. Also we define a new radical in free modules and find some characterizations of it. We prove that for $P$-SUBMODULE $N$ of a free $R$-module $F$ which $\sqrt N \subsetneqq F$, we have for any $r \in R$ and $m \in F$, $rm \in N$ implies $r \in \sqrt P$ or $m \in \sqrt N$ if and only if $\dfrac{R}{P}$-module $\dfrac{F}{N}$ is quasi torsion free.