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On Second Submodules
Abstract
Let R be a ring with identity and M be a unital right R-module. A nonzero submodule N of M is called a second submodule if N and all its nonzero homomorphic images have the same annihilator in R. The second radical of a module M is defined to be the sum of all second submodules of M. In this paper we give some results concerning second submodules and attached primes of a module and we study the second radical of a module in some cases.
In this paper, we study on the concept of strongly 2-absorbing second submodule which is a dual notion of 2-absorbing submodule and a generalization of second submodule. We give some properties and characterizations of this submodule class and investigate the relationships with second and secondary submodules.
Let R be a commutative ring with identity and Spec s (M) denote the set all second submodules of an R-module M. In this paper, we construct and study a sheaf of modules, denoted by O(N; M), on Spec s (M) equipped with the dual Zariski topology of M , where N is an R-module. We give a characterization of the sections of the sheaf O(N; M) in terms of the ideal transform module. We present some interrelations between algebraic properties of N and the sections of O(N; M). We obtain some morphisms of sheaves induced by ring and module homomorphisms.
Let R be a commutative ring with identity and S pecs(M) (resp. Min(M)) denote the set of all second (resp. minimal) submodules of a non-zero R-module M. In this paper, we investigate several properties of the subspace topology on Min(M) induced by the dual Zariski on S pecs(M) and determine some cases in which Min(M) is a max-spectral space.
In this paper, we construct some topologies on a module M by using the dual Zariski topology on a quotient module of M and define a continuous map between these topologies. Then we find a topology on M which is homeomorphic to the dual Zariski topology on a quotient module of M.
Let R be a commutative ring with identity and Spec^{s}(M) denote the set all second submodules of an R-module M. In this paper, we investigate various properties of Spec^{s}(M) with respect to different topologies. We investigate the dual Zariski topology from the point of view of seperation axioms, spectral spaces and combinatorial dimension. We establish conditions for Spec^{s}(M) to be a spectral space with respect to quasi-Zariski topology and second classical Zariski topology. We also present some conditions under which a module is cotop.
Let R be an associative ring with identity and M be a right R-module. In this paper,
we extend the de
nition of weakly second submodules of modules over commutative rings
to modules over arbitrary rings. We denote the set of all weakly second submodules of M
by Specws(M). First, we investigate some properties of weakly second submodules, and
then we introduce and study a topology on Specws(M) called the weakly second classical
Zariski topology of M. We de
ne the weakly second radical of a submodule N of M
to be the sum of all weakly second submodules of N. We determine the weakly second
radical of some modules. We de
ne the notion of weak m�-system and characterize the
weakly second radical of a submodule in terms of weak m�-systems. Then we study some
topological properties of the weakly second classical Zariski topology of M. We investigate
the irreducibility of subsets of Specws(M). We give an interrelation between irreducible
components of Specws(M) and maximal weakly second submodules of M. Finally, we
prove that if M satis
es descending chain condition on weakly second radical submodules
then Specws(M) is a spectral space.
Let R be a ring with an identity (not necessary commutative) and
let M be a left R-module. In this paper we will introduce the concept of a
comultiplication R-module and we will obtain some related results.
Uniqueness properties of coprimary decompositions of modules over non-commutative rings are presented.
In this paper the concept of the second submodule (the dual notion of prime submodule) is introduced. We show that for a finitely cogenerated (the dual notion of finitely generated) module, every non-zero submodule contains a second submodule. It is shown that, if M is a finitely generated R-module and if M is a second submodule of itself then the zero submodule of M is a prime submodule. Also the dual of this result (in some sense) is shown.
Characterizations for prime and semi-prime rings satisfying the right quotient conditions (see § 1) have been determined by A. W. Goldie in (4 and 5). A ring R is prime if and only if the right annihilator of every non-zero right ideal is zero. A natural generalization leads one to consider right R -modules having the properties that the annihilator in R of every non-zero submodule is zero and regular elements in R annihilate no non-zero elements of the module. This is the motivation for the definition of prime module in § 1.
Let R be a commutative ring (with identity). In this paper we will obtain some results concerning comultiplication R-modules. Further we state and prove a dual notion of Nakayama's lemma for finitely cogenerated modules.
Let R be a ring with identity. A unital left R-module M has the min-property provided the simple submodules of M are independent. On the other hand a left R-module M has the complete max-property provided the maximal submodules of M are completely coindependent, in other words every maximal submodule of M does not contain the intersection of the other maximal submodules of M. A semisimple module X has the min-property if and only if X does not contain distinct isomorphic simple submodules and this occurs if and only if X has the complete max-property. A left R-module M has the max-property if for every positive integer n and distinct maximal submodules L, Li (1 ≤ i ≤ n) of M. It is proved that a left R-module M has the complete max-property if and only if M has the max-property and every maximal submodule of M/Rad M is a direct summand, where Rad M denotes the radical of M, and in this case every maximal submodule of M is fully invariant. Various characterizations are given for when a module M has the max-property and when M has the complete max-property.