Ziegler's Indecomposability Criterion

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 ...

We characterize the left Noetherian rings by the existence of decompositions of left modules into direct sums of an injective submodule and a submodule containing no injective submodule except 0. We also prove that a left Noetherian ring is left hereditary iff the suspension of each left ideal (see [7]) is injective or, equivalently, the above decomposition is unique.0. Let A be an Abelian group.Then A = D B, where D is divisible and B is reduced in the sense of Kaplansky [8]; that is, B has no divisible submodule except 0. If this holds for left modules over a unitary ring R, with "injective" and "/-reduced" replacing "divisible" and "reduced", respectively, then R must be left Noetherian.The proof requires the theorem of Eilenberg-Bass (see, e.g., Theorem 20.1, p. Ill, of [2]) which characterizes left Noetherian by the requirement that direct sums of injective left modules are injective.Furthermore, if the decomposition, which will be referred to as an i-decomposition, is unique in the sense that D is the unique injective submodule of A for which A = D ffi B, with B /-reduced, then every sum of injective submodules is injective, and hence R must be left hereditary by a theorem of Matlis [9].At the same time, it will be shown that a left Noetherian ring is left hereditary iff the suspension of each left ideal is injective.

We study an $R$-module $M$ such that for every proper submodule $K$ of $M$, Hom$_R(M/K,M)\neq 0$. Such modules are called coretractable. Amini et al. investigated coretractable modules, and in this work, we obtain some new results on coretractable modules. We prove that every module over a left Artinian left H-ring is coretractable. Furthermore, we show that every module $_RM$ of finite length over a commutative ring $R$ is coretractable. Recall that we say an $R$-module $M$ is co-finite-retractable (simply CFR) whenever for every proper finitely generated submodule $K$ of $M$, Hom$_R(M/K,M)\neq 0$. We show that every CFR module is coretractable if and only if every module is coretractable. Moreover, we show that several of the module classes which we study, are equivalent over a left Noetherian ring. We prove that if $R$ is a left Noetherian ring, then $R$ is left semi-Artinian, and left H-ring if and only if every cyclic $R$-module is coretractable (CFR).

Ranks of indecomposable modules over one-dimensional rings

Injective modules and semistar operations

One of the aims of the modern representation theory is to solve classification problems for subcategories of modules over a unitary ring R. In this paper, we introduce the concept of 1-absorbing comultiplication modules and classify 1-absorbing comultiplication modules over local Dedekind domains and we study it in detail from the classification problem point of view. The main purpose of this article is to classify all those indecomposable 1-absorbing comultiplication modules with finite-dimensional top over pullback rings of two local Dedekind domains and establish a connection between the 1-absorbing comultiplication modules and the pure-injective modules over such rings. In fact, we extend the definition and results given in [17] to a more general 1-absorbing comultiplication modules case.

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

We show that if R is a left Noetherian ring, then Here, for a Noetherian left module A, len A is its ordinal valued length as defined by Gulliksen (Gulliksen, (1973). T. H. A theory of length for noetherian modules. J. Pure Appl. Algebra 3:159–170), and ⊗ is the natural product on ordinal numbers.

Criteria are given for a ring to have a left Noetherian largest left quotient ring. It is proved that each such a ring has only finitely many maximal left denominator sets. An explicit description of them is given. In particular, every left Noetherian ring has only finitely many maximal left denominator sets.

Centers and prime ideals in group algebras of polycyclic-by-finite groups

Let R be a left Noetherian ring and ZD(R) be the set of all zero-divisors of R. In this paper, it is shown that if R \\ ZD(R) is finite, then R is finite.

Let R be a left Noetherian ring, S a right Noetherian ring and Rω a Wakamatsu tilting module with S = End(Rω). We introduce the notion of the ω-torsionfree dimension of finitely generated R-modules and give some criteria for computing it. For any n ⩾ 0, we prove that l.idR(ω) = r.idS(ω) ⩽ n if and only if every finitely generated left R-module and every finitely generated right S-module have ω-torsionfree dimension at most n, if and only if every finitely generated left R-module (or right S-module) has generalized Gorenstein dimension at most n. Then some examples and applications are given.

Poor modules with no proper poor direct summands

In [1], Heller and Reiner have shown that any p-group G has infinitely many inequivalent, indecomposable rational integral representations, unless G is cyclic of order p or p2. In the present paper, we shall derive these results from a more general structural theorem in ? 2 below. This theorem states that, if 0 is any order in any algebraic number field K such that the p-group G has at least four distinct K-irreducible representations, then the group ring O[G] has infinitely many inequivalent, indecomposable torsion-free modules. From the available evidence, it seems that the converse to this statement ought to hold, i.e., that if G has three or fewer distinct K-irreducible representations, then O[G] has only finitely many distinct indecomposable torsion-free modules. Warning. We remark here once for all that any ring in this paper comes provided with an identity; any module is a left module; and the ring identity acts as the identity transformation of the module.

Let R be a left noetherian ring. We show that every complex of left R-modules has a #-injective cover. We use this result to prove that the class of DG-injective complexes is covering if and only if the ring R is regular.