A characterization of the Artinian modules.
A generalization of Mathieu subspaces to modules of associative algebras
We first propose a generalization of the notion of Mathieu subspaces of associative algebras , which was introduced recently in [Zhao W., Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra, 2010, 214(7), 1200–1216] and [Zhao W., Mathieu subspaces of associative algebras], to -modules . The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets σ(N) and τ(N) of stable...
A note on generalizations of semisimple modules
A left module over an arbitrary ring is called an -module (or an -module) if every submodule of with is a direct summand of (a supplement in, respectively) . In this paper, we investigate the various properties of -modules and -modules. We prove that is an -module if and only if , where is semisimple. We show that a finitely generated -module is semisimple. This gives us the characterization of semisimple rings in terms of -modules. We completely determine the structure of these...
A short proof of Krull's intersection theorem
Abelian group pairs having a trivial coGalois group
Torsion-free covers are considered for objects in the category Objects in the category are just maps in -Mod. For we find necessary and sufficient conditions for the coGalois group associated to a torsion-free cover, to be trivial for an object in Our results generalize those of E. Enochs and J. Rado for abelian groups.
Abelian groups which have trivial absolute coGalois group
In this article we characterize those abelian groups for which the coGalois group (associated to a torsion free cover) is equal to the identity.
Addendum to "Rings whose modules have maximal submodules".
Addendum to the author's article "Rings whose modules have maximal submodules", which appeared in Publicacions Matemàtiques 39, 1 (1995), 201-214.
Associated primes and primal decomposition of modules over commutative rings
Let R be a commutative ring and let M be an R-module. The aim of this paper is to establish an efficient decomposition of a proper submodule N of M as an intersection of primal submodules. We prove the existence of a canonical primal decomposition, , where the intersection is taken over the isolated components of N that are primal submodules having distinct and incomparable adjoint prime ideals . Using this decomposition, we prove that for ∈ Supp(M/N), the submodule N is an intersection of -primal...
Classification of the simple modules of the quantum Weyl algebra and the quantum plane
Commutative rings whose certain modules decompose into direct sums of cyclic submodules
We provide some characterizations of rings for which every (finitely generated) module belonging to a class of -modules is a direct sum of cyclic submodules. We focus on the cases, where the class is one of the following classes of modules: semiartinian modules, semi-V-modules, V-modules, coperfect modules and locally supplemented modules.
Composition-diamond lemma for modules
We investigate the relationship between the Gröbner-Shirshov bases in free associative algebras, free left modules and “double-free” left modules (that is, free modules over a free algebra). We first give Chibrikov’s Composition-Diamond lemma for modules and then we show that Kang-Lee’s Composition-Diamond lemma follows from it. We give the Gröbner-Shirshov bases for the following modules: the highest weight module over a Lie algebra , the Verma module over a Kac-Moody algebra, the Verma module...
Dually steady rings
Éléments tertiaires d'une (T)-algèbre
Energy estimate for wave equations with coefficients in some Besov type class.
Fully rigid systems of modules
Generic extensions of nilpotent k[T]-modules, monoids of partitions and constant terms of Hall polynomials
We prove that the monoid of generic extensions of finite-dimensional nilpotent k[T]-modules is isomorphic to the monoid of partitions (with addition of partitions). This gives us a simple method for computing generic extensions, by addition of partitions. Moreover we give a combinatorial algorithm that calculates the constant terms of classical Hall polynomials.
Infinite-dimensional bicomplex Hilbert spaces.
Jordan *-derivation pairs and quadratic functionals on modules over *-rings.
Kaplansky classes
Matrix rings with summand intersection property
A ring has right SIP (SSP) if the intersection (sum) of two direct summands of is also a direct summand. We show that the right SIP (SSP) is the Morita invariant property. We also prove that the trivial extension of by has SIP if and only if has SIP and for every idempotent in . Moreover, we give necessary and sufficient conditions for the generalized upper triangular matrix rings to have SIP.