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A generalization of Mathieu subspaces to modules of associative algebras

Wenhua Zhao (2010)

Open Mathematics

We first propose a generalization of the notion of Mathieu subspaces of associative algebras 𝒜 , 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 𝒜 -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

Engin Kaynar, Burcu N. Türkmen, Ergül Türkmen (2019)

Commentationes Mathematicae Universitatis Carolinae

A left module M over an arbitrary ring is called an ℛ𝒟 -module (or an ℛ𝒮 -module) if every submodule N of M with Rad ( M ) N is a direct summand of (a supplement in, respectively) M . In this paper, we investigate the various properties of ℛ𝒟 -modules and ℛ𝒮 -modules. We prove that M is an ℛ𝒟 -module if and only if M = Rad ( M ) X , where X is semisimple. We show that a finitely generated ℛ𝒮 -module is semisimple. This gives us the characterization of semisimple rings in terms of ℛ𝒮 -modules. We completely determine the structure of these...

Abelian group pairs having a trivial coGalois group

Paul Hill (2008)

Czechoslovak Mathematical Journal

Torsion-free covers are considered for objects in the category q 2 . Objects in the category q 2 are just maps in R -Mod. For R = , we find necessary and sufficient conditions for the coGalois group G ( A B ) , associated to a torsion-free cover, to be trivial for an object A B in q 2 . Our results generalize those of E. Enochs and J. Rado for abelian groups.

Associated primes and primal decomposition of modules over commutative rings

Ahmad Khojali, Reza Naghipour (2009)

Colloquium Mathematicae

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, N = N ( ) , where the intersection is taken over the isolated components N ( ) 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...

Composition-diamond lemma for modules

Yuqun Chen, Yongshan Chen, Chanyan Zhong (2010)

Czechoslovak Mathematical Journal

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 s l 2 , the Verma module over a Kac-Moody algebra, the Verma module...

Dually steady rings

Robert El Bashir, Tomáš Kepka, Jan Žemlička (2011)

Acta Universitatis Carolinae. Mathematica et Physica

Generic extensions of nilpotent k[T]-modules, monoids of partitions and constant terms of Hall polynomials

Justyna Kosakowska (2012)

Colloquium Mathematicae

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.

Kaplansky classes

Edgar E. Enochs, J. A. López-Ramos (2002)

Rendiconti del Seminario Matematico della Università di Padova

Matrix rings with summand intersection property

F. Karabacak, Adnan Tercan (2003)

Czechoslovak Mathematical Journal

A ring R has right SIP (SSP) if the intersection (sum) of two direct summands of R 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 R by M has SIP if and only if R has SIP and ( 1 - e ) M e = 0 for every idempotent e in R . Moreover, we give necessary and sufficient conditions for the generalized upper triangular matrix rings to have SIP.

Modules commuting (via Hom) with some colimits

Robert El Bashir, Tomáš Kepka, Petr Němec (2003)

Czechoslovak Mathematical Journal

For every module M we have a natural monomorphism Ψ : i I H o m R ( M , A i ) H o m R M , i I A i and we focus our attention on the case when Ψ is also an epimorphism. Some other colimits are also considered.

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