Semisimple Rings Research Articles

Additive unit representations in rings over global fields -- A survey

We give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. We focus on rings of integers in number fields and in function fields of one variable over perfect fields. The central problem is whether and how certain rings are (additively) generated by their units. In the final section we deal with matrix rings over quaternions and over Dedekind domains. Our point of view is number-theoretic whereas we do not discuss the general algebraic background. 1. The unit sum number In 1954, Zelinsky [44] proved that every endomorphism of a vector space V over a division ring D is a sum of two automorphisms, except if D = Z/2Z and dimV = 1. This was motivated by investigations of Dieudonne on Galois theory of simple and semisimple rings [7] and was probably the first result about the additive unit structure of a ring. Using the terminology of Vamos [41], we say that an element r of a ring R (with unity 1, not necessarily commutative) is k-good if r is a sum of exactly k units of R. If every element of R has this property then we call R k-good. By Zelinsky’s result, the endomorphism ring of a vector space with more than two elements is 2-good. Clearly, if R is k-good then it is also l-good for every integer l > k. Indeed, we can write any element of R as r = (r − (l − k) · 1) + (l − k) · 1, and expressing r − (l − k) · 1 as a sum of k units gives a representation of r as a sum of l units. Goldsmith, Pabst and Scott [20] defined the unit sum number u(R) of a ring R to be the minimal integer k such that R is k-good, if such an integer exists. If R is not k-good for any k then we put u(R) := ω if every element of R is a sum of units, and u(R) :=∞ if not. We use the convention k < ω <∞ for all integers k. Clearly, u(R) ≤ ω if and only if R is generated by its units. Here are some examples from [20] and [41]: • u(Z) = ω, 1991 Mathematics Subject Classification. 00-02, 11R27, 16U60.

Stratifying derived module categories

The concept of recollement is used to obtain a stratification of the derived module category of a ring which may be regarded as an analogue of a composition series for groups or modules. This analogy raises the problem whether a ‘derived’ Jordan Hölder theorem holds true; that is, are such stratifications unique up to ordering and equivalence? This is indeed the case for several classes of rings, including semi-simple rings, commutative Noetherian rings, group algebras of finite groups, and finite dimensional algebras which are piecewise hereditary.

On Fully Idempotent Modules

A submodule N of a module M is idempotent if N = Hom(M, N)N. The module M is fully idempotent if every submodule of M is idempotent. We prove that over a commutative ring, cyclic idempotent submodules of any module are direct summands. Counterexamples are given to show that this result is not true in general. It is shown that over commutative Noetherian rings, the fully idempotent modules are precisely the semisimple modules. We also show that the commutative rings over which every module is fully idempotent are exactly the semisimple rings. Idempotent submodules of free modules are characterized.

Weak-projective dimensions

In this paper, the notions of weak-projective modules and weak-projective dimension over commutative domain R are given. It is shown that over semisimple rings with weak global dimension 1, these modules are equivalent to weak-injective modules. The weak-projective dimension measures how far away a domain is from being a Prüfer domain. Several properties of these modules are also presented.

MP-injective rings and MGP-injective rings

A ring R is said to be right MP-injective if every monomorphism from a principal right ideal to R extends to an endomorphism of R. A ring R is said to be right MGP-injective if, for any 0 ≠ a ∈ R, there exists a positive integer n such that a n ≠ 0 and every monomorphism from a n R to R extends to R. We shall study characterizations and properties of these two classes of rings. Some interesting results on these rings are obtained. In particular, conditions under which right MGP-injective rings are semisimple artinian rings, von Neumann regular rings, and QF-rings are given.

Direct sums of injective semimodules and direct products of projective semimodules over semirings

We prove that if the direct sum of a family of semimodules over a semiring S is an injective semimodule or if the direct product of a family of semimodules over S is a projective semimodule, then the cardinality of the subfamily consisting of all semimodules which are not modules is strictly less than the cardinality of S. As a consequence, we obtain semiring analogs of well-known characterizations of classical semisimple, quasi-Frobenius, and one-sided Noetherian rings.

ON THE OSOFSKY–SMITH THEOREM

AbstractWe recall a version of the Osofsky–Smith theorem in the context of a Grothendieck category and derive several consequences of this result. For example, it is deduced that every locally finitely generated Grothendieck category with a family of completely injective finitely generated generators is semi-simple. We also discuss the torsion-theoretic version of the classical Osofsky theorem which characterizes semi-simple rings as those rings whose every cyclic module is injective.

Partial Skew Polynomial Rings Over Semisimple Artinian Rings

Let R be a semisimple Artinian ring with a partial action α of ℤ on R, and let R[x; α] be the partial skew polynomial ring. By the classification of the set E of all minimal central idempotents in R into three different types, a complete description of the prime radical of R[x; α] is given. Moreover, it is shown that any nonzero prime ideal of R[x; α] is maximal and is either principal or idempotent. In the case where α is of finite type, it is shown that R[x; α] is a semiprime hereditary ring.

On the endofiniteness of a key module over pure semisimple rings

Let R R be a left pure semisimple ring such that there are no non-zero homomorphisms from preinjective modules to non-preinjective indecomposable modules in R R -mod, and let W W be the left key R R -module; i.e., W W is the direct sum of all non-isomorphic non-preinjective indecomposable direct summands of products of preinjective left R R -modules. We show that if the module W W is endofinite, then R R is a ring of finite representation type. This settles a question considered in [L. Angeleri Hügel, A key module over pure-semisimple hereditary rings, J. Algebra 307 (2007), 361-376] for hereditary rings.

Quasi-pseudo Principally Injective Modules

In this paper, the concept of quasi-pseudo principally injective modules is introduced and a characterization of commutative semi-simple rings is given in terms of quasi-pseudo principally injective modules. An example of pseudo M-p-injective module which is not M-pseudo injective is given.

Pure Hereditary Rings

In this article, the notion of pure hereditary rings is introduced as a nontrivial generalization of the notion of both pure semisimple rings and of hereditary rings. Some properties and examples of pure hereditary rings are given. For a positive integer n, let 𝒫 n be the class of all left R-modules with projective dimension ≤ n. It is shown that over a left pure hereditary ring is a perfect and hereditary cotorsion pair, which gives a partial positive answer to a question of Göbel and Trlifaj. The modules in the class are also studied. Special attention is paid to the case n = 1.

WHEN COTORSION MODULES ARE PURE INJECTIVE

We characterize rings over which every cotorsion module is pure injective (Xu rings) in terms of certain descending chain conditions and the Ziegler spectrum, which renders the classes of von Neumann regular rings and of pure semisimple rings as two possible extremes. As preparation, descriptions of pure projective and Mittag–Leffler preenvelopes with respect to so-called definable subcategories and of pure generation for such are derived, which may be of interest on their own. Infinitary axiomatizations lead to coherence results previously known for the special case of flat modules. Along with pseudoflat modules we introduce quasiflat modules, which arise naturally in the model-theoretic and the category-theoretic contexts.

Model companion and model completion of theories of rings

Extending the language of rings to include predicates for Jacobson radical relations, we show that the theory of regular rings defined by Carson, Lipshitz and Saracino is the model completion of the theory of semisimple rings. Removing the requirement on the Jacobson radical (reduced to {0}), we prove that the theory of rings with no nilpotents does not admit a model companion relative to this augmented language.

Hermitian Morita Theory and Unitary Groups of Modules over Semisimple Artinian Rings

This article investigates isomorphisms between certain subgroups of the projective unitary groups of hermitian modules over semisimple Artinian rings with anti-structures. These subgroups contain the commutator subgroups of the projective unitary groups. Specifically, the article provides conditions under which these isomorphisms are induced by and the underlying rings are connected by hermitian Morita equivalences (HMEs). This article introduces the hyperbolic length of a module as well as the concept of generalized hyperbolic modules over simple Artinian rings, and over semisimple Artinian rings with anti-structures. The article shows that the stated isomorphisms are induced by HMEs if all the following conditions hold: (a) the hermitian forms are nonsingular and trace valued; (b) the modules in question are generalized hyperbolic; (c) the hyperbolic length equals three or is greater than or equal to five (hyperbolic length is greater than or equal to five in the semisimple case). Significantly, the condition of the hyperbolic length of a module greater than or equal to m is satisfied by a set of modules larger than or equal to those satisfying the condition of the Witt index of the module greater than or equal to m.

Semisimple rings on completely decomposable Abelian groups

An Abelian group is said to be semisimple if it is the additive group of some semisimple associative ring. The problem of describing semisimple groups was formulated by Beaumont and Lawver; later this problem was reduced to the case of reduced groups. In this paper, we describe semisimple groups in the class of countable completely decomposable groups.

The Gabriel–Roiter Measure for Right Pure Semisimple Rings

We show how the Gabriel–Roiter measure, introduced by Ringel in (Bull Sci Math 129:726–748, 2005 and Contemp Math 406:105–135, 2006), applies to indecomposable modules of finite length over right pure semisimple rings, and in particular to the study of the open problem whether any right pure semisimple ring is of finite representation type.

Gorenstein flatness and injectivity over Gorenstein rings

Let R be a Gorenstein ring. We prove that if I is an ideal of R such that R/I is a semi-simple ring, then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical. In addition, we prove that if R → S is a homomorphism of rings and SE is an injective cogenerator for the category of left S-modules, then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical. We also give some applications of these results.

PURE SEMISIMPLE FINITELY ACCESSIBLE CATEGORIES AND HERZOG'S CRITERION

Let [Formula: see text] be a finitely accessible category with products, and assume that its symmetric category [Formula: see text] is also finitely accessible and pure semisimple. We study necessary and sufficient conditions in both categories for [Formula: see text] (and hence [Formula: see text]) to be of locally finite representation type. In particular, we obtain a generalization of Herzog's criterion for finite representation type of left pure semisimple and right artinian rings. As an application, we prove that a left pure semisimple ring R with enough idempotents which has a self-duality is of locally finite representation type if and only if it is left locally finite.

Preinjective modules over pure semisimple rings

For a left pure semisimple ring R , it is shown that the local duality establishes a bijection between the preinjective left R -modules and the preprojective right R -modules, and any preinjective left R -module is the source of a left almost split morphism. Moreover, if there are no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in R -mod, the direct sum of all non-preinjective indecomposable direct summands of products of preinjective left R -modules is a finitely generated product-complete module. This generalizes a recent theorem of Angeleri Hügel [L. Angeleri Hügel, A key module over pure-semisimple hereditary rings, J. Algebra 307 (2007) 361–376] for hereditary rings.

When every self-small module is finitely generated

The aim of this paper is to give necessary and sufficient conditions for rings for which every right self-small module is finitely generated. It is proved that: semi-simple rings, commutative perfect rings and right nonsingular Σ-extending rings have this property; a right nonsingular semi-prime ring has the property if and only if it is semi-simple; a commutative noetherian ring has the property if and only if it is artinian.