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.

Please click on PDF to view the abstract

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.

Let R be a commutative ring with identity. The pur-pose of this paper is to introduce AND study two classes of modules over R, called Max-injective AND Max-STRONGLY top modules AND explore some of their basic properties. Our concern is to extend some properties of X-injective AND STRONGLY top modules to these classes of modules AND obtain some related results.

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.

Let $R= \bigoplus_{g \in G} R_g$ be a $G-$graded commutative ring with identity, $I$ be a graded ideal AND let $M$ a $G-$graded unitary $R$-module, where $G$ is a semigroup with identity $e$. We introduce graded $I-$PRIME ideals (SUBMODULEs) as a generalizations of the classical notions of PRIME ideals (SUBMODULEs). We show that the new notions inherite the basic properties of the classical ones. In particular, we investigate the localization theory of these two concepts. We prove that for a faithfull flat module $F$, a graded SUBMODULE $P$ of $M$ is $I-$PRIME if AND only if $F \otimes P$ is graded $I-$PRIME SUBMODULE of $F \otimes M$. As an application, for finitely generated graded module $M$ over Noetherian graded ring $R$, the completion of graded $I-$PRIME SUBMODULEs is $I-$PRIME SUBMODULE.