DECOMPOSITION OF ARTINIAN ALTERNATIVE RINGS

Throughout this paper the word “ring” will mean an associative ring which need not have an identity element. There are Artinian rings which are not Noetherian, for example C(p∞) with zero multiplication. These are the only such rings in that an Artinian ring R is Noetherian if and only if R contains no subgroups of type C(p∞) [1, p. 285]. However, a certain class of Artinian rings is Noetherian. A famous theorem of C. Hopkins states that an Artinian ring with an identity element is Noetherian [3, p. 69]. The proofs of these theorems involve the method of “factoring through the nilpotent Jacobson radical of the ring”. In this paper we state necessary and sufficient conditions for an Artinian ring (and an Artinian module) to be Noetherian. Our proof avoids the concept of the Jacobson radical and depends primarily upon the concept of the length of a composition series. As a corollary we obtain the result of Hopkins.

In this paper we prove a new characterisation of hereditary PI rings, namely we show that a Noetherian, but not Artinian, PI ringR that is an order in an Artinian ring splits into a direct sum of an Artinian ring of finite representation type and hereditary semiprime rings if and only if all its proper Artinian factor rings are of finite representation type. We also show, through examples, that the above characterisation does not hold for some more general settings.

Every indecomposable module over a generalized uniserial ring is uniserial, hence local. This motivates one to study rings R satisfying the condition (*): R is a right artinian ring such that every finitely generated, indecomposable right R-module is local. The rings R satisfying (*) have been recently studied by Singh and Al-Bleahed (2004), they have proved some results giving the structure of local right R-modules. In this paper some more structure theorems for local right R-modules are proved. Examples given in this paper show that a rich class of rings satisfying condition (*) can be constructed. Using these results, it is proved that any ring R satisfying (*) is such that mod-R is of finite representation type. It follows from a theorem by Ringel and Tachikawa that any right R-module is a direct sum of local modules. If M is right module over a right artinian ring such that any finitely generated submodule of any homomorphic image of M is a direct sum of local modules, it is proved that it is a direct sum of local modules. This provides an alternative proof for that any right module over a right artinian ring R satisfying (*) is a direct sum of local modules.

Some Pathology for Radicals in Non-Associative Near-Rings

Commuting maps and identities with inverses on alternative division rings

The polynomial ring over a Goldie ring need not be a Goldie ring

The author develops the structure theory of alternative and Jordan compact rings, and proves that in such rings the quasiregular radical in the strong sense is topologically nilpotent, and the quotient ring by this radical is isomorphic to a full direct sum of finite prime fields with the Tikhonov topology.Bibliography: 11 titles.

A module M is defined to be a CS module, if every submodule of M is essential in a direct summand of M. In this paper we show, among other results, that for a right nonsingular ring R, all direct sums of CS right R-modules are CS if and only if R is a right artinian ring and every indecomposable infective right R-module has length at most two.KeywordsDirect SummandDivision RingInjective ModuleEndomorphism RingSemiprimary RingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A module is uniserial in case its submodules are linearly ordered by inclusion. A ring R is left (right) serial if it is a direct sum of uniserial left (right) R-modules. A ring R is serial if it is both left and right serial. It is well known that for artinian rings the property of being serial is equivalent to the finitely generated modules being a direct sum of uniserial modules [8]. Results along this line have been generalized to more arbitrary rings [6], [13].This article is concerned with investigating rings whose indecomposable injective modules are uniserial. The following question is considered which was first posed in [4]. If an artinian ring R has all indecomposable injective modules uniserial, does this imply that R is serial? The answer is yes if R is a finite dimensional algebra over a field. In this paper it is shown, provided R modulo its radical is commutative, that R has every left indecomposable injective uniserial implies that R is right serial.

Let R be an artinian ring. A family, ℳ, of isomorphism types of R-modules of finite length is said to be canonical if every R-module of finite length is a direct sum of modules whose isomorphism types are in ℳ. In this paper we show that ℳ is canonical if the following conditions are simultaneously satisfied: (a) ℳ contains the isomorphism type of every simple R-module; (b) ℳ has a preorder with the property that every nonempty subfamily of ℳ with a common bound on the lengths of its members has a smallest type; (c) if M is a nonsplit extension of a module of isomorphism type II1 by a module of isomorphism type II2, with II1, II2 in ℳ, then M contains a submodule whose type II3 is in ℳ and II1 does not precede II3. We use this result to give another proof of Kronecker's theorem on canonical pairs of matrices under equivalence. If R is a tame hereditary finite-dimensional algebra we show that there is a preorder on the family of isomorphism types of indecomposable R-modules of finite length that satisfies Conditions (b) and (c).

Let R be a left and right artinian ring with identity, and / the Jacobson radical of R. In [4], M. Harada has considered a left serial ring R satisfying a condition (*, 2) that every maximal submodule of a direct sum of any two hollow modules is also a direct sum of hollow modules, and characterized such a ring by the structure of eR for each primitive idempotent e. Further it has been shown that the condition (*, 2) is equivalent to saying that every factor module of ejQ)eR is a direct sum of hollow modules for every primitive idempotent e. Modifying this, we here consider the following condition on a projective indecomposable right module eR over a ring R.

It is well known that two basic rings are Morita equivalent if and only if they are isomorphic.Here it is shown that local uniserial rings are stably equivalent in the sense of M. Auslander and I. Reiten in case they are isomorphic modulo certain powers of their radicals.In particular, two commutative local uniserial rings of Loewy length n are stably equivalent if and only if they are isomorphic modulo the [/i/2]th power of their radicals.In a series of papers [2]-[6], M. Auslander and I. Reiten have studied categories of modules modulo projectives and the related notion of stable equivalence of rings.We recall their terminology.For a ring A, mod A denotes the category of finitely generated left A-modules.For M, N G mod A, PA(M, N) is the subgroup of HomA(M, N) consisting of the homomorphisms /: M -> N that factor through projectives.The additive category of finitely generated left A-modules modulo projectives is mod A. Its objects are the same as those of mod A, and its morphisms are the members of the factor groups HomA(A/, N) = HomA(A/, N)/PA(M, N).The category modF A is the full subcategory of mod A whose objects have no projective direct summands, and mod,, A is the corresponding subcategory of mod A. If A is a left artinian ring then mod,, A mod A ( denotes equivalence of categories) [3, p. 247].Two rings A and A' are said to be (left) stably equivalent in case mod A mod A'.Stable equivalence is a relaxing of the notion of Morita equivalence.So since artinian rings are Morita equivalent iff their basic rings are isomorphic (see [1, 27]), one is led to study the relative structure of stably equivalent rings.For example, although Zp2 and Zp[x]/(x2) are not Morita equivalent, they are stably equivalent [2].See also [6] and [7] regarding stable equivalence of serial rings.Here we consider stable equivalence of uniserial rings.A uniserial ring is one that is Morita equivalent to a direct sum of local uniserial rings.A ring A with J = J (A) and Loewy length t? is a local uniserial ring in case A > J > > J"~x > J" = 0 is a. complete list of the submodules of both AA and AA.The indecomposable modules over A are then A/J, ..., A/J"~x, A, and PA(A/J', A/Jj) consists of those maps that factor through A. In what

Local rings whose modules are almost serial

Let R be an associative ring with identity.Then the left socle of R is a direct summand of R as a right ϋNmodule if and only if it is projective as a left jR-module and contains no infinite sets of orthogonal idempotents.This implies, for example, that a ring with finitely generated left socle and no nilpotent minimal left ideals is a ring direct sum of a semisimple artinian ring and a ring with zero left socle.

Abstract. Let R be a ring. A right R -module M is PS -injective if every R -homomorphism f : aR ! M for every principally small right ideal aR can be extendedto R ! M . We investigate, in this paper, rings whose simple singular modules are PS -injective. New characterizations of semiprimitive rings and semisimple Artinian rings aregiven. 1. IntroductionThroughout this paper, R is an associative ring with identity and all modulesare unitary. The Jacobson radical of R is denoted by J ( R ) and the right singularideal is denoted by Z ( R R ). For a ∈ R , l ( a )(resp. r ( a )) denote the left (resp. right)annihilator of a in R . For the usual notations we refer the reader to [3], [7] and [10].A right ideal I of R is called small if for every proper right ideal K of R , K + I = R . A right R -module M is right PS -injective if every R -homomorphism f : aR → M for every principally small right ideal aR can be extended to R → M (see [13]). The ring R is said to be right PS -injective if R R