On modules with the Kulikov property and pure semisimple modules and rings (original) (raw)
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Indecomposable modules over right pure semisimple rings
Monatshefte f�r Mathematik, 1988
The aim of this paper is to prove the following result. If A is a fight pure semisimple ring, then it satisfies one of the two following statements: (a) For any positive integer n, there are at most finitely many indecomposable right modules of length n; or (b) There is an infinite number of integers d such that, for each d, A has infinitely many indecomposable right modules of length d. The result is derived with the aid of ultraproduct-technique.
Some properties of purely simple Kronecker modules, I
Journal of Pure and Applied Algebra, 1983
Let K be an algebraically closed field. A K2-system is a pair of K-vector spaces (V, W) together with a K-bilinear map from K' x V to W. The category of systems is equivalent to the category of right modules over some K-algebra, R. Most of the concepts in the theory of modules over the polynomial ring K[<] have analogues in Mod-R. Unlike the purely simple K(C]-modules, which are easily described, purely simple R-modules are quite complex. If M is a purely simple R-module of finite rank n then any submodule of M of rank less than n is finite-dimensional. The following corollaries are derived from this fact: 1, Every non-zero endomorphism of M is manic. 2. Every torsion-free quotient of M is purely simple. 3. An ascending union of purely simple R-modules of increasing rank is not purely simple. It is also shown that a large class of torsion-free rank one modules can occur as the quotient of a purely simple system of rank n, n any positive integer. Moreover. starting from a purely simple system another purely simple module M' of the same rank is constructed and M' is shown to be both a submodule of M and a submodule of a rank 1 torsion-free system. Since the category of right R-modules is a full subcategory of right S-modules, where S is any finite-dimensional hereditary algebra of tame type, the paper provides a way of constructing infinite-dimensional indecomposable S-modules.
Semi-Endosimple Modules and Some Applications
2009
An R-module is called semi-endosimple if it has no proper fully invariant essential submodules. For a quasi-projective retractable module M R we show that M is finitely generated semi-endosimple if and only if the endomorphism ring of M is a finite direct sum of simple rings. For an arbitrary module M , conditions equivalent to the semi-endosimplicity of its quasi-injective hull are found. As consequences of these results, new characterizations of V-rings, right Noetherian V-rings and strongly semiprime rings are obtained. As such, a hereditary left Noetherian ring R is a finite direct sum of simple Noetherian right V-rings if and only if all finitely generated right R-modules are semi-endosimple.
Virtually semisimple modules and a generalization of the Wedderburn-Artin theorem
Communications in Algebra
By any measure, semisimple modules form one of the most important classes of modules and play a distinguished role in the module theory and its applications. One of the most fundamental results in this area is the Wedderburn-Artin theorem. In this paper, we establish natural generalizations of semisimple modules and give a generalization of the Wedderburn-Artin theorem. We study modules in which every submodule is isomorphic to a direct summand and name them virtually semisimple modules. A module R M is called completely virtually semisimple if each submodules of M is a virtually semisimple module. A ring R is then called left (completely) virtually semisimple if R R is a left (compleatly) virtually semisimple R-module. Among other things, we give several characterizations of left (completely) virtually semisimple rings. For instance, it is shown that a ring R is left completely virtually semisimple if and only if R ∼ = k i=1 M ni (D i) where k, n 1 , ..., n k ∈ N and each D i is a principal left ideal domain. Moreover, the integers k, n 1 , ..., n k and the principal left ideal domains D 1 , ..., D k are uniquely determined (up to isomorphism) by R.
arXiv: Rings and Algebras, 2020
Recently, in a series of papers "simple" versions of direct-injective and directprojective modules have been investigated (see, [4], [12], [13]). These modules are termed as "simple-direct-injective" and "simple-direct-projective", respectively. In this paper, we give a complete characterization of the aforementioned modules over the ring of integers and over semilocal rings. The ring is semilocal if and only if every right module with zero Jacobson radical is simple-direct-projective. The rings whose simple-direct-injective right modules are simple-direct-projective are fully characterized. These are exactly the left perfect right Hrings. The rings whose simple-direct-projective right modules are simple-direct-injective are right max-rings. For a commutative Noetherian ring, we prove that simple-direct-projective modules are simple-direct-injective if and only if simple-direct-injective modules are simpledirect-projective if and only if the ring is Artinian. Various closure properties and some classes of modules that are simple-direct-injective (resp. projective) are given.
On semi-projective modules and their endomorphism rings
Asian-European Journal of Mathematics, 2017
This paper provides the several homological characterization of perfect rings and semi-simple rings in terms of semi-projective modules. We investigate whether Hopkins–Levitzki Theorem extend to semi-projective module i.e. whether there exists an artinian semi-projective module which are noetherian. Unfortunately, the answer we have is negative; counter example is given. However, it is shown that the answer is positive for certain large classes of semi-projective modules in Proposition 2.26. We have discussed the summand intersection property, summand sum property for semi-projective modules. Apart from this, we have introduced the idea of [Formula: see text]-hollow modules, also several necessary and sufficient conditions are established when the endomorphism rings of a semi-projective modules is a local ring.
A note on indecomposable modules
Rendiconti del Circolo Matematico di Palermo, 1988
In this note we study rings having only a finite number of non isomorphic uniform modules with non zero socle. It is proved that a commutative ring with this property is a direct sum of a finite ring and a ring of finite representation type. In the non commutative case we show that most P.I. rings having only a finite number of non isomorphic modules with non zero socle are in fact artinlan.
Proceedings of the Japan Academy, Series A, Mathematical Sciences, 1987
Introduction. Let R be a fixed (not necessarily commutative) ring. Throughout this nte, we are concerned with left R-modules M, A, H, Like in'Goldie [1], we shall use the following terminology. A non-zero submodule K of M is called essential in M (or M is an essential extension