Semisimple Artinian Research Articles

Modules and abelian groups with a bounded domain of injectivity

In this work, impecunious modules are introduced as modules whose injectivity domains are contained in the class of all pure-split modules. This notion gives a generalization of both poor modules and pure-injectively poor modules. Properties involving impecunious modules as well as examples that show the relations between impecunious modules, poor modules and pure-injectively poor modules are given. Rings over which every module is impecunious are right pure-semisimple. A commutative ring over which there is a projective semisimple impecunious module is proved to be semisimple artinian. Moreover, the characterization of impecunious abelian groups is given. It states that an abelian group [Formula: see text] is impecunious if and only if for every prime integer [Formula: see text], [Formula: see text] has a direct summand isomorphic to [Formula: see text] for some positive integer [Formula: see text]. Consequently, an example of an impecunious abelian group which is neither poor nor pure-injectively poor is given so that the generalization defined is proper.

Divisibility on chains of submodules

ABSTRACTAn R-module M is said to satisfy ACCd (resp. DCCd) on submodules if for every ascending (resp. descending) chain {Mi} of submodules of M, (resp. ) for some for i≫0. A nonzero module with ACCd or DCCd on submodules contains an essential submodule which is a direct sum of uniform submodules almost all noetherian. We show that if M is a finitely generated self-projective and self-injective R-module with ACCd or DCCd on submodules, then M is a finite direct sum of uniform submodules. It is shown that a regular right self-injective ring with ACCd or DCCd on right ideals must be semisimple artinian. We also prove that if M is a nonzero nonsingular module over a right noetherian ring and E(M)(ℕ) satisfies ACCd or DCCd on submodules, then M is semisimple. Next we consider some conditions for modules with ACCd (resp. DCCd) on submodules to satisfy ACC (resp. DCC) on some families of prime submodules. Finally, we show that a commutative ring with DCCd on ideals has dimension at most 1.

Rings whose cyclics are C3-modules

It is well known that if every cyclic right module over a ring is injective, then the ring is semisimple artinian. This classical theorem of Osofsky promoted a considerable interest in the rings whose cyclics satisfy a certain generalized injectivity condition, such as being quasi-injective, continuous, quasi-continuous, or [Formula: see text]. Here we carry out a study of the rings whose cyclic modules are [Formula: see text]-modules. The motivation is the observation that a ring [Formula: see text] is semisimple artinian if and only if every [Formula: see text] -generated right [Formula: see text]-module is a [Formula: see text]-module. Many basic properties are obtained for the rings whose cyclics are [Formula: see text]-modules, and some structure theorems are proved. For instance, it is proved that a semiperfect ring has all cyclics [Formula: see text]-modules if and only if it is a direct product of a semisimple artinian ring and finitely many local rings, and that a right self-injective regular ring has all cyclics [Formula: see text]-modules if and only if it is a direct product of a semisimple artinian ring, a strongly regular ring and a [Formula: see text] matrix ring over a strongly regular ring. Applications to the rings whose [Formula: see text]-generated modules are [Formula: see text] -modules, and the rings whose cyclics are ADS or quasi-continuous are addressed.

Generalized Injectivity and Approximations

Injective, pure-injective, and fp-injective modules are well known to provide for approximations in the category Mod-R for an arbitrary ring R. We prove that this fails for many other generalizations of injectivity: the C1, C2, C3, quasi-continuous, continuous, and quasi-injective modules. We show that, except for the class of all C1-modules, each of the latter classes provides for approximations only when it coincides with the injectives (for quasi-injective modules, this forces R to be a right noetherian V-ring; in the other cases, R even has to be semisimple artinian). The class of all C1-modules over a right noetherian ring R is (pre)enveloping iff R is a certain right artinian ring of Loewy length ≤2; in this case, however, R may have an arbitrary representation type.

Rings with each right ideal automorphism-invariant

In this paper, we study rings having the property that every right ideal is automorphism-invariant. Such rings are called right a-rings. It is shown that (1) a right a-ring is a direct sum of a square-full semisimple artinian ring and a right square-free ring, (2) a ring R is semisimple artinian if and only if the matrix ring Mn(R) is a right a-ring for some n>1, (3) every right a-ring is stably-finite, (4) a right a-ring is von Neumann regular if and only if it is semiprime, and (5) a prime right a-ring is simple artinian. We also describe the structure of an indecomposable right artinian right non-singular right a-ring as a triangular matrix ring of certain block matrices.

Indecomposable decomposition and couniserial dimension

Dimensions like Gelfand, Krull, Goldie have an intrinsic role in the study of theory of rings and modules. They provide useful technical tools for studying their structure. We define and study couniserial dimension for modules. Couniserial dimension is a measure of how far a module deviates from being uniform. Despite their different objectives, it turns out that there are certain common properties between the couniserial dimension and Krull dimension. Among others, each module having such a dimension contains a uniform submodule and has finite uniform dimension. Like all dimensions, this is an ordinal valued invariant. Every module of finite length has couniserial dimension and its value lies between the uniform dimension and the length of the module. Modules with countable couniserial dimension are shown to possess indecomposable decomposition. In particular, a von Neumann regular ring with countable couniserial dimension is semisimple artinian. If the maximal right quotient ring of a semiprime right non-singular ring \(R\) has a couniserial dimension as an \(R\)-module, then \(R\) is a semiprime right Goldie ring. As one of the applications, it follows that all right \(R\)-modules have couniserial dimension if and only if \(R\) is a semisimple artinian ring.

Modules Whose Endomorphism Rings are Von Neumann Regular

Abelian groups whose endomorphism rings are von Neumann regular have been extensively investigated in the literature. In this paper, we study modules whose endomorphism rings are von Neumann regular, which we call endoregular modules. We provide characterizations of endoregular modules and investigate their properties. Some classes of rings R are characterized in terms of endoregular R-modules. It is shown that a direct summand of an endoregular module inherits the property, while a direct sum of endoregular modules does not. Necessary and sufficient conditions for a finite direct sum of endoregular modules to be an endoregular module are provided. As a special case, modules whose endomorphism rings are semisimple artinian are characterized. We provide a precise description of an indecomposable endoregular module over an arbitrary commutative ring. A structure theorem for extending an endoregular abelian group is also provided.

RINGS OVER WHICH CYCLICS ARE DIRECT SUMS OF PROJECTIVE AND CS OR NOETHERIAN

AbstractR is called a right WV-ring if each simple right R-module is injective relative to proper cyclics. If R is a right WV-ring, then R is right uniform or a right V-ring. It is shown that a right WV-ring R is right noetherian if and only if each right cyclic module is a direct sum of a projective module and a CS (complements are summands, a.k.a. ‘extending modules’) or noetherian module. For a finitely generated module M with projective socle over a V-ring R such that every subfactor of M is a direct sum of a projective module and a CS or noetherian module, we show M = X ⊕ T, where X is semisimple and T is noetherian with zero socle. In the case where M = R, we get R = S ⊕ T, where S is a semisimple artinian ring and T is a direct sum of right noetherian simple rings with zero socle. In addition, if R is a von Neumann regular ring, then it is semisimple artinian.

Relating properties of a ring with properties of matrix rings coming from Ornstein dual pairs

In this paper we study the ring-theoretic classification of intermediate rings, Eαβ(R) of infinite matrix rings, RCFMα(R)⊆Eαβ(R)⊆RFMα(R), from Ornstein's dual pairs (see [D. Ornstein, Dual vector spaces, Ann. Math. 69 (1959) 520–534; A. del Rio, J.J. Simón, Intermediate rings between matrix rings and Ornstein dual pairs, Arch. Math. (Basel) 75 (2000) 256–263]) in case R is semisimple artinian, semiprimary or left or right perfect. To do this, we develop a technique of decomposing an infinite matrix as “infinite sum of submatrices of less size.” Then, we show that R is a semisimple artinian ring if and only if Eαβ(R) is a von Neumann regular ring. We then describe the lattice of finitely generated left ideals, and the lattice of two-sided ideals of Eαβ(R) by equivalence of idempotents; that is, in classical terms. We also describe the Jacobson radical of Eαβ(R) for an arbitrary ring R, and then we show, among other results, that R is left perfect if and only if Eαβ(R) is a semiregular ring.

A RESULT ON SEMI-ARTINIAN RINGS

AbstractIt was shown by Huynh and Rizvi that a ring $R$ is semisimple artinian if and only if every continuous right $R$-module is injective. However, a characterization of rings, over which every finitely generated continuous right module is injective, has been left open. In this note we give a partial solution for this question. Namely, we show that for a right semi-artinian ring $R$, every finitely generated continuous right $R$-module is injective if and only if all simple right $R$-modules are injective.AMS 2000 Mathematics subject classification: Primary 16D50. Secondary 16P20; 16P60

Semisimple strongly graded rings

is semisimple (meaning in this paper semisimple artinian) has been studied byseveral authors. The most classical result is Maschke’s Theorem for group rings.For crossed products over fields there is a satisfactory answer given by Aljadeffand Robinson [3]. Another partial answer for skew group rings was given byAlfaro et al. [1]. A reduction of the problem to crossed products over divisionrings was first given by Jespers and Okninski [10]and a more constructiveversion´was given by Haefner and del Rio [8]. So, in order to give a complete answer tothe problem there is still a gap between crossed products over division rings andcrossed products over fields. The first aim of the paper is to fill this gap, showingthat the semisimplicity question for crossed products over division rings reducesto the same question for crossed products over fields. In Theorem 3.2 we makethis reductionand then in Theorem3.3 we put togetherall the piecesof the puzzle.A strongly

ABELIAN GROUPS WITH SEMI-LOCAL ENDOMORPHISM RING

ABSTRACT The abelian groups with semi-local endomorphism ring are characterized, with one exception: the infinite rank torsion-free ones. A ring is called semi-local if is semisimple artinian. In his authoritative book on infinite abelian groups, Laszlo Fuchs, the leading expert on this topic, asks: “For which abelian groups is the endomorphism ring semi-local?” ([1], Problem 84). This problem has been open for 26 years. Modules whose endomorphism ring is semi-local have been investigated by several authors (see, e.g.,[3] and the literature listed there). The main result of this paper is the following theorem: Theorem. Let be an abelian group with endomorphism ring and torsion subgroup . Then: is semi-local if and only if is finitely generated and is a direct sum of form , where is a torsion-free subgroup of such that is semi-local. If is divisible, then is semi-local if and only if has finite rank. If is torsion-free reduced of finite rank, then is semi-local if and only if , for all but finitely many pr...

Rings with all modules residually finite

Define a ringA to be RRF (resp. LRF) if every right (resp. left) A-module is residually finite. Refer to A as an RF ring if it is simultaneously RRF and LRF. The present paper is devoted to the study of the structure of RRF (resp. LRF) rings. We show that all finite rings are RF. IfA is semiprimary, we show thatA is RRF ⇔A is finite ⇔A is LRF. We prove that being RRF (resp. LRF) is a Morita invariant property. All boolean rings are RF. There are other infinite strongly regular rings which are RF. IfA/J(A) is of bounded index andA does not contain any infinite family of orthogonal idempotents we prove: (i) A an RRF ring ⇔ A right perfect andA/J(A) finite (henceA/J(A) finite semisimple artinian). (ii) A an LRF ring ⇔ A left perfect andA/J(A) finite

On the rank of the first radical layer of ap-class group of an algebraic number field

Letpbe a prime number. LetMbe a finite Galois extension of a finite algebraic number fieldk. Suppose thatMcontains a primitive pth root of unity and that thep-Sylow subgroup of the Galois groupG= Gal(M/k) is normal. LetKbe the intermediate field corresponding to thep-Sylow subgroup. Let= Gal(K/k). Thep-class groupCofMis a module over the group ringZpG, whereZpis the ring ofp-adic integers. LetJbe the Jacobson radical ofZpG. C/JCis a module over a semisimple artinian ringFp. We study multiplicity of an irreducible representation Φ apperaring inC/JCand prove a formula giving this multiplicity partially. As application to this formula, we study a cyclotomic fieldMsuch that the minus part ofCis cyclic as aZpG-module and a CM-fieldMsuch that the plus part ofCvanishes for oddp.To show the formula, we apply theory of central extensions of algebraic number field and study global and local Kummer duality between the genus group and the Kummer radical for the genus field with respect toM/K.

Semisimple artinian graded rings

Semisimple artinian graded rings

Mackey Functors with Chain Conditions

LetAbe a Green functor for a finite groupGand letMbe a leftA-module.Mis callednoetherian (artinian)if the lattice of all submodules ofMsatisfies ACC (respectively DCC).Mis calledtotally noetherian (totally artinian)ifMXis noetherian (respectively artinian) for every finiteG-setX. In this paper it is shown that the totally noetherian (totally artinian) leftA-modules behave in the same way as the classical noetherian (artinian) modules. The left noetherian (left artinian) Green functors are less well behaved but some properties from classical algebra still hold. It is shown that Cohen's theorem holds in this setting. Our main result shows that if a commutative Green functorAis artinian, then the ringsA(H)/Jac(A(H)) are semisimple artinian for all subgroupsHofG.

An approach to Boyle's conjecture

A ring R is called a right QI-ring if every quasi-injective right R-module is injective. The well-known Boyle's Conjecture states that any right QI-ring is right hereditary. In this paper we show that if every continuous right module over a ring R is injective, then R is semisimple artinian. In fact, if every singular continuous right R-module satisfying the restricted semisimple condition is injective, then R is right hereditary. Moreover, in this case, every singular right R-module is injective.

Semisimple rings of quotients in Morita contexts

Let ( R, R U S , S V R , S) be a Morita context with the trace ideals I in R and J in S, τ a Gabriel topology containing I on R-Mod, and τ′ the corresponding Gabriel topology containing J on S-Mod. Necessary and sufficient conditions on τ and R U are given in order that the ring of quotients Ŝ of S with respect to τ′ be semisimple artinian, simple artinian or a division ring respectively. Special cases include earlier works.

Additive rank functions in noetherian rings

Let R be a ring with unit element. A function 1, that assigns to each finitely generated R-module M a nonnegative integer 1(M) such that J(M) = A(N) + 1(L) for each short exact sequence 0 + N--f M + L --f 0 of finitely right R-modules is called an additive rank function. Such a function is constant on isomorphism classes of finitely generated right R-modules, so that 1(M/cr(M)) = 0 whenever CI: A4 -+ M is an injective endomorphism of M. When R is semisimple artinian, then an additive rank function arises very naturally by defining A(M) = length(M), the number of simple direct summands in a decomposition of M. In 1964, Goldie [2] adapted this idea to finitely generated modules over a right noetherian ring, the resulting additive rank function is presently called the reduced rank. The concept lay dormant for over 10 years until Goldie started using it in 1976, providing new and strikingly elegant proofs of several important results of ring theory. Some of these appear in [3], others are included in [ 11. Since then, the reduced rank has become a widely used tool, particularly in the theory of noncommutative noetherian rings. We recall the definition of the reduced rank p(M) of a finitely generated right R-module M over a right noetherian ring R. If N denotes the nilpotent radical of R then R/N has a classical semisimple artinian quotient ring Q(R/N). For a finitely generated right R/N-module M set pRIN(M) = length(M@.,, Q(R/N)). Since R,NQ(R/N) is flat, pRIN is an additive rank function for finitely generated right R/N-modules. For a finitely generated right R-module M set

Auslander Algebras as Quasi-Hereditary Algebras

The notion of a quasi-hereditary algebra has been introduced by E. Cline, B. Parshall and L. Scott [7,2,5] in order to describe the so-called highest weight categories arising in the representation theory of Lie algebras and algebraic groups. Quasi-hereditary algebras are defined by the existence of a suitable chain of ideals, and the finite dimensional hereditary algebras are typical examples. In [3], also finite dimensional algebras of global dimension 2 are shown to be quasi-hereditary. Thus, the Auslander algebras are quasi-hereditary. Recall that the Auslander algebras A can be constructed in the following way. Let R be a representation-finite finite dimensional algebra; then A is the endomorphism algebra End(Affl), where M is a finite dimensional /{-module such that every indecomposable /{-module is isomorphic to a direct summand of M. We are going to introduce the notion of a splitting filtration on the class of all indecomposable /{-modules and show that in this way we obtain a heredity chain of ideals of A (see the definition below). Usually, there exist many splitting filtrations for a given R. Examples of splitting filtrations can be obtained from the Rojter measure, used by A. V. Rojter in his proof of the first Brauer-Thrall conjecture [6], or from the preprojective and preinjective partitions, introduced by M. Auslander and S. Smalo in [1]. Instead of dealing with finite dimensional algebras, we shall consider, more generally, semiprimary rings. Recall that an associative ring A with 1 is called semiprimary provided that its Jacobson radical TV is nilpotent and A/N is semisimple artinian. We say that an ideal J of A is a heredity ideal of A if P = J, JNJ = 0 and /, considered as a right ideal, is a projective ^-module. Following [2], a semiprimary ring A is said to be quasi-hereditary provided that there exists a chain