Let $R$ be a commutative ring and let $M$ be an $R$-module. In this paper, we introduce the dual notion of fuzzy prime (that is, fuzzy second) submodules of $M$ and explore some of the basic properties of this class of submodules. We say a non-zero fuzzy submodule $\mu$ of $M$ is fuzzy second if for each $r \in R$, we have $1_r. \mu = \mu$ or $1_r. \mu = 1_\theta$. It is shown that the fuzzy second submodules is a proper subclass of the fuzzy coprimary submodules.

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.

In this paper, the concept of fuzzy visible submodules which is a new type of fuzzy submodules has been introduced. Some results and characterizations of fuzzy visible are established namely the homomorphic image of the fuzzy visible submodule, the sum of two fuzzy visible submodules. The relation between fuzzy visible submodule and its submodules. Also, the fuzzy quotient modules in sense of fuzzy visible have been presented. We prove that the intersection of a collection of fuzzy visible submodules are visible submodules and the converse is not true. Also, we define the strong cancellation fuzzy modules and we established some results of it with respect to fuzzy visible submodules. Many other properties we study in fuzzy visible submodules.