The Large Condition for Rings with Krull Dimension

We show that a simple Noetherian ring of finite global dimension and Krull dimension one is Monta equivalent to a domain.In [4] is given an example of a simple Noetherian ring that is not Morita equivalent to a domain.Now the ring in question has infinite global dimension but Krull dimension one.In contrast, we show here that if R is a simple Noetherian ring of finite global dimension and Krull dimension one, then R is Morita equivalent to a domain.This is a corollary of Theorem 1, which gives a generalisation of this result to simple rings of arbitrary finite Krull dimension.Throughout this note all rings will contain an identity and all modules will be unitary.We will denote the uniform dimension of a module M by ud(M) and the projective dimension by pd(M).The left Krull dimension of a ring R

Let A be an integral domain, X an analytic indeterminate over A and I a proper ideal (not necessarily prime) of A. In this paper, we study the ring [Formula: see text] First, we study the prime spectrum of R. We prove that the contraction map: Spec (A[[X]]) → Spec (R); Q ↦ Q ∩ R induces a homeomorphism, for the Zariski's topologies, from {Q ∈ Spec (A[[X]]) | XI[[X]] ⊈ Q} onto {P ∈ Spec (R) | XI[[X]] ⊈ P}. If P ∈ Spec (R) is such that XI[[X]] ⊆ P then there exists p ∈ Spec (A) such that P = p + XI[[X]]. Next, we study the Krull dimension of R. We give a necessary condition for R to be of finite Krull dimension. In particular, if R is of finite dimension then I must be an SFT ideal of A. Then we determine bounds for dim (R). Examples are given to indicate the sharpness of the results. In case I is a maximal ideal of A and A is either a Noetherian ring, SFT Prüfer domain or A[[X]] is catenarian and I SFT, we establish that dim (R) = dim (A[[X]]) = dim (A) + 1. Finally, we examine the possible transfer of the LFD property and the catenarity between the rings A, A[[X]] and R in case I is a maximal ideal of A.

Totally acyclic complexes

Let R be an associative ring with identity, and let denote the category of unital left R-modules. The Walkers [6] raised the question of characterizing the maximal torsion radicals of , and showed that if R is commutative and Noetherian, then there is a one-to-one correspondence between maximal torsion radicals and minimal prime ideals of R [6, Theorem 1.29]. Popescu announced [5, Theorem 2.5] that the result remains valid for commutative rings with Gabriel dimension (in the terminology of [2]). Theorem 4.6 below shows that the result holds for rings (not necessarily commutative) with Krull dimension on either the left or right, extending the previous theorem for right Noetherian rings which appeared in [1].

SynopsisFor each ring R, we construct a topological space pt (R) which includes as a subspace both the classical spectrum specR and the torsion theoretic spectrum R-sp. For many rings (e.g. rings with Krull dimension), spec R is a retract of pt (R) and the retraction map θ generalizes the Gabriel correspondence for noetherian rings. There is a natural decomposition theory on MOD-R which extends the Goldman theory in the same way that the tertiary theory extends the primary theory. The map θ provides a direct comparison between this new decomposition theory and the tertiary theory. The space pt (R) is closely connected with the lattice of hereditary torsion theories on R, and for fully bounded (not necessarily noetherian) R, this connection is very tight.

We introduce and study lattice-finite Noetherian rings and show that they form a onedimensional analogue of representation-finite Artinian rings. We prove that every lattice-finite Noetherian ring R has Krull dimension ≼ 1, and that R modulo its Artinian radical is an order in a semi-simple ring. Our main result states that maximal overorders of R exist and have to be Asano orders, while they need not be fully bounded. This will be achieved by means of an idempotent ideal I(R), an invariant or R which is new even for classical orders R. This ideal satisfies I(R) = R whenever R is maximal.

We define an ordinal valued length for Noetherian modules which extends the usual definition of composition series length for finite length modules. Though originally defined by Gulliksen[7] in the 1970s, this extension has seldom been used in subsequent research. Despite this neglect, we will show that the ordinal valued length is a quite natural measure of the size of a Noetherian module, and has advantages over more familiar measures such as uniform dimension, Krull dimension, and reduced rank. We will also demonstrate how some familiar properties of left Noetherian rings can be proved efficiently using length and the arithmetic properties of ordinal numbers.

ABSTRACTWe extend the definition of a piecewise Noetherian ring to the noncommutative case, and investigate various properties of such rings. In particular, we show that a ring with Krull dimension is piecewise Noetherian. Certain fully bounded piecewise Noetherian rings have Gabriel dimension and exhibit the Gabriel correspondence between prime ideals and indecomposable injective modules.

Let R be a commutative ring and let ℒ(σ) be a Gabriel filter of R such that R is σ-noetherian. We discuss the decomposition of the σ 1-torsion submodule of a σ-torsionfree R-module and characterize the σ l-injectivity of σ-closed R-modules through the σ m-injectivity of modules over noetherian local rings (S, m). As an application, we obtain new criteria to determine injectivity of modules over noetherian rings, of finite Krull dimension, and Krull domains.

This paper contributes several results to the analytic theory of ideals. The main new concept is that of a prereduction (together with the closely related type-prereduction): if R is a ring and I (proper subset) are proper ideals in R, then A is called a prereduction of I in case A is not a reduction of I, but each ideal between A and I is a reduction of I. It is shown that each non-nilpotent proper ideal in a Noetherian ring has at least one prereduction, and a number of the basic properties of the ideals in the set of all prereductions of I are proved. Also, if I is a non-nilpotent proper ideal in a local (Noetherian) ring (R, M), then:There is a natural one-to-one correspondence between the set of the equivalence classes of the type-prereductions of I and the set of the maximal relevant ideals in the Rees ring of I (that is, the homogeneous prime ideals Q in the ring (t an indeterminate) such that and the Krull dimension of is equal to one). is the set of maximal elements in for all positive integers n, and, for each for all large integers m.A complete description is given of the prereductions and the elements in for each ideal I that is generated by analytically independent elements.If R/M is algebraically closed and I is a non-nilpotent and non-principal ideal in R, then there is a natural one-to-one correspondence between the sets and and for all positive integer n.

Throughout this note, R stands for a ring with identity, and all modules are unital R-modules. In this note, we ask when the class of all finitely generated reflexive (torsionless) left modules is closed under extensions. This is not always the case, even if R is left and right noetherian (cf. Bass [2, Lemma 3.2] and [3, p. 12]). As will be shown, in case R is a commutat ive noetherian local domain of Krull dimension two, R is a CohenMacaulay ring whenever the class of the finitely generated reflexive modules is closed under extensions. Our main aim is to show that if R is a left and right noetherian ring then both the class of all finitely generated torsionless left modules and the class of all finitely generated reflexive left modules are closed under extensions if and only if for i = 1, 2 the functor Ext , 1 (Ext~ ( , R), R) vanishes on the finitely generated right modules. Note that the latter condition is satisfied if R is a commutat ive Gorenstein ring (see Bass [3]). We will show also that if R is a left and right noetherian ring with a minimal injective resolution 0 ~ RR ~ Eo ~ E1 , . . . such that weak dim E l < i + 1 for i = 0, 1 then both the class of finitely generated torsionless left modules and the class of finitely generated reflexive left modules are closed under extensions. In the proof, it will be shown that if R is a left and right noetherian ring with inj dim RR _--< 2 then the class of the finitely generated reflexive left modules is closed under extensions. On the other hand, even if R is a left and right noetherian ring with inj dim RR = inj dim R R < 2, the class of all finitely generated torsionless left modules is not necessarily closed under extensions. In what follows, we denote by ( )* both the R-dual functors, and for a given module M we denote by eu: M ~ M** the usual evaluation map and by E(M) the injective envelope of M. Recall that a module M is said to be torsionless if eu is a monomorph i sm and to be reflexive if eu is an isomorphism. Note that a module M is torsionless if and only if it is cogenerated by R. We denote by mod R (resp. mod R ~ the category of the finitely presented left (resp. right) modules. Note that if R is left noetherian then every finitely generated left module is finitely presented.

For a commutative ring A we study here the Grothendieck group, K0A, of finitely generated projective A-modules, and the Picard group, Pic (A), of projective modules of rank one (under (?A). Writing koA for the elements of rank zero in KOA, there is an epimorphism det: K0A Pic (A) defined by the rth exterior power of a projective module of rank r. Our original objective was to calculate KJ(Zw), where w is a finitely generated abelian group. We do this in ? 8. The bulk of the paper is a rather general discussion of the techniques, some already familiar in other contexts, which we use to make this calculation. Our procedure is to decompose w as w0 x T, with w0 a finite group and T free abelian, and then to write Zw= A[T] where A = Zw0. Then A is a noetherian ring of (Krull) dimension one. For such an A we show (Theorem 7.8) that det: ko(A[T]) Pic (A[T]) is an isomorphism. In ?9 we prove also a non-stable analogue of this for T of rank <2. In ?8 we show (Theorem 8.1) that

We study modules which are characteristic submodules in their injective hulls. The author is supported by the Russian Foundation for Basic Research, project 08-01- 00693-a All rings below are assumed to be associative and with nonzero identity ele- ment; all modules are assumed to be unitary. A moduleM is said to be Noetherian if every properly ascending chain of submodules in M is finite. Expressions of the form 'A is a Noetherian ring' mean that AA and AA are Noetherian modules. A moduleM is said to be injective with respect to the moduleX or X-injective if for any submodule X1 in X, every homomorphismX1 → M may be extended to a homomorphismX → M. For a ring A, an A-module M is said to be injective if M is injective with respect to any A-module. A submodule M of the module E is said to be essential if M ∩ E1 6 0 for any nonzero submodule E1 in E. If E is an injective module and M is an essential submodule in E then E is called the injective hull of the module M. The injective hull is unique up to isomorphism. A submoduleM of the module E is called a characteristic submodule if f(M) ⊆ M for every automorphismf of the module E. In the papers (3), (8), and (10), a module M is said to be automorphism- invariant if M is a characteristic submodule of the injective hull of M. It is easy to verify that a module M is an automorphism-invariant if and only if M is a charac- teristic submodule of some injective module; see Remark 6 below. Remark 1. In (3, Theorem 16), it is proved that a module M is an automorphism-invariant module if and only if M is a pseudo-injective module, i.e., for any submodule X in M , every monomorphism X → M may be exten- ded to an endomorphism of the moduleM . For example, see (6), (11), (3) about pseudo-injective modules. Remark 2. It is clear that every injective module is an automorphism-invariant module. The converse is not true, since every simple Abelian group is a finite automorphism-invariant non-injective module over the ring of integers Z.

A commutative ring R is said to be fragmented if each nonunit of R is divisible by all positive integral powers of some corresponding nonunit of R. It is shown that each fragmented ring which contains a nonunit non-zero-divisor has (Krull) dimension ∞. We consider the interplay between fragmented rings and both the atomic and the antimatter rings. After developing some results concerning idempotents and nilpotents in fragmented rings, along with some relevant examples, we use the “fragmented” and “locally fragmented” concepts to obtain new characterisations of zero-dimensional rings, von Neumann regular rings, finite products of fields, and fields.

Let A be a commutative ring and n ≤ 3 a positive integer. In this paper, we consider unimodular rows (f1(x),..., f n (x)) over A[x]. We prove that, if the row of the leading coefficients of f i (x) is unimodular over A and a ɛ A, then there exists β ɛ E n (A[x]) such that (f1(x),...,f n (x))β = (f 1 (a),...,f n (a)). Also, if A is a Noetherian ring with finite Krull dimension and the row of leading coefficients satisfies the same condition, then we give a bound for the length of β in terms of elementary transvections.