### A characterization of the Artinian modules.

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We first propose a generalization of the notion of Mathieu subspaces of associative algebras $$\mathcal{A}$$ , 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 $$\mathcal{A}$$ -modules $$\mathcal{M}$$ . 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 left module $M$ over an arbitrary ring is called an $\mathrm{\mathcal{R}\mathcal{D}}$-module (or an $\mathrm{\mathcal{R}\mathcal{S}}$-module) if every submodule $N$ of $M$ with $\mathrm{Rad}\left(M\right)\subseteq N$ is a direct summand of (a supplement in, respectively) $M$. In this paper, we investigate the various properties of $\mathrm{\mathcal{R}\mathcal{D}}$-modules and $\mathrm{\mathcal{R}\mathcal{S}}$-modules. We prove that $M$ is an $\mathrm{\mathcal{R}\mathcal{D}}$-module if and only if $M=\mathrm{Rad}\left(M\right)\oplus X$, where $X$ is semisimple. We show that a finitely generated $\mathrm{\mathcal{R}\mathcal{S}}$-module is semisimple. This gives us the characterization of semisimple rings in terms of $\mathrm{\mathcal{R}\mathcal{S}}$-modules. We completely determine the structure of these...

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 $R=\mathbb{Z},$ we find necessary and sufficient conditions for the coGalois group $G(A\u27f6B),$ associated to a torsion-free cover, to be trivial for an object $A\u27f6B$ in ${q}_{2}.$ Our results generalize those of E. Enochs and J. Rado for abelian groups.

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 the author's article "Rings whose modules have maximal submodules", which appeared in Publicacions Matemàtiques 39, 1 (1995), 201-214.

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, $N={\bigcap}_{}{N}_{\left(\right)}$, where the intersection is taken over the isolated components ${N}_{\left(\right)}$ 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...

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...

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.

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 $(1-e)Me=0$ for every idempotent $e$ in $R$. Moreover, we give necessary and sufficient conditions for the generalized upper triangular matrix rings to have SIP.

For every module $M$ we have a natural monomorphism $$\Psi :\coprod _{i\in I}{\mathrm{H}om}_{R}(M,{A}_{i})\to {\mathrm{H}om}_{R}\left(M,\coprod _{i\in I}{A}_{i}\right)$$ and we focus our attention on the case when $\Psi $ is also an epimorphism. Some other colimits are also considered.