A new characterisation of hereditary pi rings

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.

Special Quasi-triads and Integral Group Rings of Finite Representation Type, I

Quasi-Artin species and rings of finite representation type

A right [Formula: see text]-module [Formula: see text] is called to satisfy condition [Formula: see text] if, for every [Formula: see text] and [Formula: see text], there exists [Formula: see text] such that [Formula: see text] and if [Formula: see text] is a direct summand of [Formula: see text], then [Formula: see text] is a direct summand of [Formula: see text]. In this paper, we give some properties of rings and modules to satisfy condition [Formula: see text]. Moreover, their connections with von Neumann regular rings, Hereditary rings, Noetherian rings and (semi)artinian rings are addressed.

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

On the ℓ-DLIPs of codes over finite commutative rings

It is proved that a right distributive semiprime PI ringA is a left distributive ring and for each elementx ∈A there is a positive integern such thatx n A=Ax n . We describe both right distributive right Noetherian rings algebraic over the center of the ring and right distributive left Noetherian PI rings. We also characterize rings all of whose Pierce stalks are right chain right Artin rings.

In this paper we study some classes of rings which have a finite lattice of preradicals. We characterize commutative rings with this condition as finite representation type rings, i.e., artinian principal ideal rings. In general, it is easy to see that the lattice of preradicals of a left pure semisimple ring is a set, but it may be infinite. In fact, for a finite dimensional path algebra Λ over an algebraically closed field we prove that Λ-pr is finite if and only if its quiver is a disjoint union of finite quivers of type An; hence there are path algebras of finite representation type such that its lattice of preradicals is an infinite set. As an example, we describe the lattice of preradicals over Λ = kQ when Q is of type An and it has the canonical orientation

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.

In this paper a number of theorems are obtained on decompositions of Artinian alternative rings, generalizing analogous theorems of Szász, Šneĭdmjuller, Kertész et al. for associative Artinian rings. For example, it is shown that every alternative Artinian ring decomposes into a direct sum of torsion parts and a torsion free ideal. With several additional restrictions, we obtain a decomposition of any alternative Artinian ring into a direct sum of an ideal and a left ideal, allowing a rather detailed description. A description of hereditarily Artinian alternative rings is given. Bibliography: 16 titles.

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