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 an R-module.Let Ψ : S(M) →S(M) ∪ ∅ be a function, where S(M) denotes the set ofall submodules of M. The main purpose of this paper is to introduce andinvestigate the notion of strongly -2-absorbing second submodules of M as ageneralization of strongly 2-absorbing second and -second submodules of M.

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.

Let R be a commutative ring with identity, S be a mul-tiplicatively closed subset of R, and let M be an R-module. In this paper, we introduce and investigate some properties of the notion of S-2-absorbing second submodules of M as a generalization of S-second submodules and strongly 2-absorbing second submodules of M. Also, we obtain some results concerning S-2-absorbing submodules of M.

‎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.

In the present paper, the analytical solution for an exact second order fuzzy initial value problem under generalized Hukuhara differentiability is obtained. First the solution of first order linear fuzzy differential equation under generalized Hukuhara differentiability is investigated using integration factor methods in two cases. The second based on the type of generalized Hukuhara differentiability, the analytical solution of an exact second order fuzzy initial value problem is derived in two senses of differentiability. Also, several properties for generalized Hukuhara differentiability are obtained on the topics, such as, the univariate fuzzy chain rules, fuzzy integration by parts, among others. At the end, some illustrative examples are presented and these examples show the behavior of the ‎solutions.