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.

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In this paper, we introduce the concepts of STRONGLY 2-absorbing primary ideals (resp., SUBMODULEs) AND STRONGLY 2-absorbing ideals (resp., SUBMODULEs) as generalizations of STRONGLY PRIME ideals. Furthermore, we investigate some basic properties of these classes of ideals.

Let M be a module over a commutative ring R. We continue our study of STRONGLY annihilating SUBMODULE graph SAG(M) introduced in [11]. In addition to providing the more properties of this graph, we introduce the subgraph SAG∗ (M) of SAG(M) AND compare the properties of SAG∗ (M) with SAG(M) AND AG(M) (the annihilating SUBMODULE graph of M introduced in [4]).

Let R be a commutative integral domain with quotient field K AND let P be a nonzero STRONGLY PRIME ideal of R. We give several characterizations of such ideals. It is shown that (P : P) is a valuation domain with the unique maximal ideal P. We also study when P-1 is a ring. In fact, it is proved that P-1 = (P : P) if AND only if P is not invertible. Furthermore, if P is invertible, then R = (P : P) AND P is a principal ideal of R.

It is well known that the sum of two z-ideals in C(X) is either C(X) or a z-ideal. The main aim of this paper is to study the sum of STRONGLY z-ideals in RL, the ring of real-valued continuous functions on a frame L. For every ideal I in RL, we introduce the biggest STRONGLY z-ideal included in I AND the smallest STRONGLY z-ideal containing I, denoted by Isz AND Isz, respectively. We study some properties of Isz AND Isz: Also, it is observed that the sum of any family of minimal PRIME ideals in the ring RL is either RL or a PRIME STRONGLY z-ideal in RL. In particular, we show that the sum of two PRIME ideals in RL which are not chains is a PRIME STRONGLY z-ideal.

The SUBMODULEs with the property of the title (a SUBMODULE N of an R -module M is called STRONGLY dense in M, denoted by N£sd M, if for any index set I, ΠIN£d ΠIM) are introduced AND fully investigated. It is shown that for each SUBMODULEN of M there exists the smallest subsetD′ ÍM such that N+D′ is a STRONGLY dense SUBMODULE of M AND D′ ∩ N=0. We also introduce a class of modules in which the two concepts of strong essentiality AND strong density coincide. It is also shown that for any module M, dense SUBMODULEs in M are STRONGLY dense if AND only if M £sd ~E (M), where ~E (M) is the rational hull of M. It is proved that R has no STRONGLY dense left ideal if AND only if no nonzeroelement of every cyclic R -module M has a STRONGLY dense annihilator in R. Finally, some properties AND new concepts related to strong density are studied.