Dual perspectively decomposable modules

Square-free modules with the exchange property

If Λ \Lambda is an artin algebra there is a partition of ind ⁡ Λ \operatorname {ind} \Lambda , the category of indecomposable finitely generated Λ \Lambda -modules, ind ⁡ Λ = ∪ i ⩾ 0 P _ _ i \operatorname {ind} \Lambda = { \cup _{i \geqslant 0}}{\underline {\underline {P}}_i} , called the preprojective partition. We show that P _ _ i \underline {\underline {P}}_i can be easily constructed for hereditary artin algebras, if P _ _ i − 1 \underline {\underline {P}}_{i - 1} is known: A A is in P _ _ i \underline {\underline {P}}_i if and only if A A is not in P _ _ i − 1 \underline {\underline {P}}_{i - 1} and there is an irreducible map B → A B \to A , where B B is in P _ _ i − 1 \underline {\underline {P}}_{i - 1} .

A module M is called dual-square-free (DSF) if M has no proper submodules A and B with M = A + B and M / A ≅ M / B . The class of DSF-modules is closed under direct summands and homomorphic images, and a module M is distributive iff every submodule of M is DSF. Every DSF-module M is Dedekind finite, and if M satisfies the finite-exchange property, then M satisfies the substitution property and its endomorphism ring has stable range 1. Moreover, a DSF-module M has the finite exchange property iff M is clean, iff M has the full exchange property. Finally, maximal submodules of DSF-modules are shown to be fully-invariant, in particular a ring R is DSF as a right R-module iff R is right quasi-duo (i.e. every maximal right ideal is two-sided). The later result provides a new perspective on the symmetry question of quasi-duo rings.

In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

Stable modules and a theorem of Camillo and Yu

Quasi-projective modules with the finite exchange property

Twisted generalized Weyl algebras (TGWAs) A(R,σ,t) are defined over a base ring R by parameters σ and t, where σ is an n-tuple of automorphisms, and t is an n-tuple of elements in the center of R. We show that, for fixed R and σ, there is a natural algebra map \(A(R,\sigma ,tt^{\prime })\to A(R,\sigma ,t)\otimes _{R} A(R,\sigma ,t^{\prime })\). This gives a tensor product operation on modules, inducing a ring structure on the direct sum (over all t) of the Grothendieck groups of the categories of weight modules for A(R,σ,t). We give presentations of these Grothendieck rings for n = 1,2, when \(R=\mathbb {C}[z]\). As a consequence, for n = 1, any indecomposable module for a TGWA can be written as a tensor product of indecomposable modules over the usual Weyl algebra. In particular, any finite-dimensional simple module over \(\mathfrak {sl}_{2}\) is a tensor product of two Weyl algebra modules.

In this paper, we introduce and study the properties of chain conditions on M-cyclic submodules defined as M-Noetherian and M-Artinian modules and rings. We prove that the property of being iso-M-Noetherian (iso-M-Artinian) is inherited by submodules, quotient modules, and finite sums as well as finite direct sums. Finally, we prove the Hilbert basis theorem for iso-M-Noetherian (iso-M-Artinian) and verify that the property of modules of being M-Noetherian (M-Artinian) and iso-M-Noetherian (iso-M-Artinian) are Morita invariant.

Let A ?= kQ/I be a basic and connected finite dimension algebra over closed field k. In this note show that in case B = A[M] is a tame one-point extension of a tame concealed algebra A by an indecomposable module M, then the trivial extension T(B) = B ? DB is tame if and only if the module M is regular.

A module [Formula: see text] is called coseparable ([Formula: see text]-coseparable) if for every submodule [Formula: see text] of [Formula: see text] such that [Formula: see text] is finitely generated ([Formula: see text] is simple), there exists a direct summand [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is finitely generated. In this paper, we show that free modules are coseparable. We also investigate whether or not the ([Formula: see text]-)coseparability is stable under taking submodules, factor modules, direct summands, direct sums and direct products. We show that a finite direct sum of coseparable modules is not, in general, coseparable. But the class of [Formula: see text]-coseparable modules is closed under finite direct sums. Moreover, it is shown that the class of coseparable modules over noetherian rings is closed under finite direct sums. A characterization of coseparable modules over noetherian rings is provided. It is also shown that every lifting (H-supplemented) module is coseparable ([Formula: see text]-coseparable).

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.

This chapter is the beginning of an examination of algebras that are more general than the semisimple algebras. Our strategy is to generalize the process that led to the Wedderburn Structure Theorem. The appropriate substitutes for the basic building blocks of that theory—simple modules—are indecomposable modules. These modules are introduced in this chapter. The analogue of Schur’s lemma provides a characterization of indecomposable modules in terms of their endomorphism algebras. The main result of the chapter is the Krull-Schmidt Theorem. It leads to the conclusion that finitely generated modules over an Artinian algebra decompose uniquely into direct sums of indecomposable modules. In short, the results of Chapter 2 have close analogues in the theory of modules over Artinian algebras.

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

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.

Let N be a normal subgroup of a group G. An N-module Q is called G-stable provided that Q is equivalent to the twist Q g of Q by g, for every g ∈ G. If the action of N on Q extends to an action of G on Q, then Q is obviously G-stable, but the converse need not hold. A famous conjecture in the modular representation theory of reductive algebraic groups G asserts that the (obviously G-stable) projective indecomposable modules (PIMs) Q for the Frobenius kernels G r (r ≥ 1) of G have a G-module structure. It is sometimes just as useful (for a general module Q) to know that a finite direct sum Q ⊕ n of Q has a compatible G-module structure. In this paper, this property is called numerical stability. In recent work (Parshall and Scott, Adv Math 226:2065–2088, 2011), the authors established numerical stability in the special case of PIMs. We provide in this paper a more general context for that result, working in the context of k-group schemes and a suitable version of G-stability, called strong G-stability. Among our results here is the determination of necessary and sufficient conditions for the existence of a compatible G-module structure on a strongly G-stable N-module, in the form of a cohomological obstruction which must be trivial precisely when the G-module structure exists. Our main result is achieved by giving an approach to killing the obstruction by tensoring with certain finite dimensional G/N-modules.