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, a novel risk-based, two-objective (technical and economical) optimal reactive POWER dispatch method in a wind-integrated POWER system is proposed which is more consistent with operational criteria.  The technical objective includes the minimization of the new voltage instability risk index. The economical objective includes cost minimization of reactive POWER generation and active POWER loss. The proposed voltage instability risk employs a hybrid possibilistic (Delphi-Fuzzy)-probabilistic approach that takes into consideration the operator’s experience, the wind speed and demand forecast uncertainties when quantifying the risk index. The decision variables are the reactive POWER resources of the system. To solve the problem, the modified multi-objective particle swarm optimization algorithm with sine and cosine acceleration coefficients is utilized. The method is implemented on the modified IEEE 30-bus system. The proposed method is compared with those in the previously published literature, and the results confirm that the proposed risk index is better at estimating the voltage instability risk of the system, especially in cases with severe impact and low probability. In addition, according to the simulation results compared to typical security-based planning, the proposed risk-based planning may increase the security and economy of the system due to better utilization of system resources.

‎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.

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.

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