Weak Dimension Research Articles

WEAK DIMENSION AND CHAIN-WEAK DIMENSION OF ORDERED SETS

In this paper, we define the weak dimension and the chain-weak dimension of an ordered set by using weak orders and chain-weak orders, respectively, as realizers. First, we prove that if P is not a weak order, then the weak dimension of P is the same as the dimension of P. Next, we determine the chain-weak dimension of the product of k-element chains. Finally, we prove some properties of chain-weak dimension which hold for dimension.

The weak dimensions of Gaussian rings

We provide necessary and sufficient conditions for a Gaussian ring $R$ to be semihereditary, or more generally, of $w.dimR\leq 1$. Investigating the weak global dimension of a Gaussian coherent ring $R$, we show that the only values $w.dimR$ may take are $0,1$ and $\infty$; but that $fP.dimR$ is always at most one. In particular, we conclude that a Gaussian coherent ring $R$ is either Von Neumann regular, or semihereditary, or non-regular of $w.dimR=\infty$.

Column Ranks and Their Preservers of Matrices Over Max Algebra

The column rank of an m by n matrix A over max algebra is the weak dimension of the column space of A. We compare the column rank with rank of matrices over max algebra. We also characterize the linear operators which preserve the column rank of matrices over max algebra.

Bezout and semihereditary power series rings

If R is a commutative ring then the polynomial ring R[x] is semihereditary if and only if R[x] is a Bezout ring, if and only if R is von Neumann regular. A good characterization of rings such that the polynomial ring is either semihereditary or Bezout is not known in the non-commutative setting. Goursaud and Valette proved in [9] that R[x] right semihereditary implies R von Neumann regular but the converse fails to be true; for any field k, R =∏n∈NMn(k) is a counterexample. Interesting results around this problem can be found in works by Goursaud and Valette [9], Menal [15], Moncasi and Goodearl [8], and Dicks and Schofield [5]. The characterization of semihereditary rings of power series over commutative rings was done by Brewer, Rutter and Watkins in [4]; they proved that R❏x❑ is Bezout if and only if R is an א0-injective von Neumann regular ring. Their motivation was to study the behavior of the weak dimension under the power series construction, characterizing nonzero Bezout power series rings as the ones of weak dimension 1. Moreover, they showed that R❏x❑ is semihereditary if and only if R is an א0-injective, א0-complete von Neumann regular ring. These results are collected in the book by Brewer [3]. In this paper we study Bezout and semihereditary power series rings in the noncommutative setting. In Theorem 1.8 we show that the power series ring over a left א0-injective von Neumann regular rings is right Bezout. We stress that this result makes a difference with the polynomial case, since it shows that for power series rings the results in the commutative case can be, at least, partially extended. The main ingredient in the proof is to find a special type of generators for principal ideals, cf. Proposition 1.5 and Corollary 1.7. In Corollary 1.9 we use this description of principal ideals to show that power series rings over non-zero left א0-injective von Neumann regular rings have weak dimension 1.

Bounds for the Homological Dimensions of Pullbacks

A new upper bound on the global dimension of a pullback ring is given, and Scrivanti's upper bound on the weak dimension of a pullback ring is generalized to the noncommutative case. Bibliography: 8 titles.

Rank preservers of matrices over max algebra

The max algebra consists of the nonnegative real numbers equipped with two binary operations, maximization and multiplication. We consider the semimodules over max algebra and study the properties of the weak basis and weak dimension of the semi-modules. Moreover, we obtain the characterizations of those linear operators that preserve rank of matrices over max-algebra.

Brand Research and the Global Conundrum

THE TOOL IS called the Brand Asset® Valuator1. It measures brand differentiation, relevance, esteem, and knowledge, qualities that can be combined in certain ways to assess brand strength and brand stature. Referring to many examples, David Redhill summarizes how the technique works and, more important, how, with related studies, it can be used on a country‐by‐country basis to pinpoint the successful and weak dimensions of any brand.

An equivalence induced by Ext and Tor applied to the finitistic weak dimension of coherent rings

Let R be a ring, see below for other notation. The functor categories (mod-R, Ab) and ((R-mod)op, Ab) have received considerable attention since the 1960s. The first of these has achieved prominence in the model theory of modules and most particularly in the investigation of the representation theory of Artinian algebras. Both [11, Chapter 12] and [8] contain accounts of the use (mod-R, Ab) may be put to in the model theoretic setting, and Auslander's review, [1], details the application of (mod-R, Ab) to the study of Artinian algebras. The category ((R-mod)op, Ab) has been less fully exploited. Much work, however, has been devoted to the study of the transpose functor between R-mod and mod-R. Warfield's paper, [13], describes this for semiperfect rings, and this duality is an essential component in the construction of almost split sequences over Artinian algebras, see [4]. In comparison, the general case has been neglected. This paper seeks to remedy this situation, giving a concrete description of the resulting equivalence between (mod-R, Ab) and ((R-mod)op, Ab) for an arbitrary ring R.

Dubrovin valuation properties of skew group rings and crossed products

In this paper some conditions for a skew group ring or a crossed product to have finite weak global dimension are given.Using these results we obtain some necessary conditions and some sufficient conditions for a skew group ring or a crossed product to be a Dubrovin valuation ring.If R*G is a skew group ring, where the coefficient ring R is a commutative ring and G is a finite group, then we prove that the conditions we obtained become necessary and sufficient conditions.In particular, if R is a commutative valuation ring, then R*G is a Dubrovin valuation ring if and only if G T=<1>,where G T is the inertial group of R.

On the global and weak dimensions of pullbacks of non-commutative rings

On the global and weak dimensions of pullbacks of non-commutative rings

Rings of weak dimension one and syzygetic ideals

We prove that rings of weak dimension one are the rings with all (three-generated) ideals syzygetic. This leads to a characterization of these rings in terms of the André-Quillen homology.

Total quality management experiences in some New South Wales manufacturing companies

Total quality management is viewed as a continuous improvement system (CIS). Quantitative indicators to measure organizational and structural features of enterprises on their way to becoming CISs are suggested. These input indicators are related to attributes pertaining to three dimensions: quality management, quality information and human resources. To identify eventual patterns representing specific models of improvement paths, we use these indicators to analyze a small but random sample of enterprises (selected from a sampling population of enterprises that had formally embarked on a total quality management programme) and detect their strong and weak dimensions. The findings provide evidence that the input indicators (whose performance statistic is expressed here in terms of a signal-to-noise ratio) have a positive causal effect on some quality improvement benefits (output indicators) and, accordingly, that a relatively high and balanced distribution of input efforts of a company over all the defined attributes can be considered the best path towards improvement. A quantitative approach to analyze managers' attitudes towards some quality-related topics using the repertory grid technique is also presented.

On Crossed Products of Hopf Algebras

Let $B = A{\# _\sigma }H$ denote a crossed product of the associative algebra A with the Hopf algebra H. We investigate the weak dimension and the global dimension of B and show that ${\text {wdim}}\;B \leq {\text {wdim}}\;H + {\text {wdim}}\;A$ and ${\text {l.gldim}} \; B \leq {\text {r.gldim}} \; H + {\text {l.gldim}} \; A$.

On crossed products of Hopf algebras

Let B = A # σ H B = A{\# _\sigma }H denote a crossed product of the associative algebra A with the Hopf algebra H. We investigate the weak dimension and the global dimension of B and show that wdim B ≤ wdim H + wdim A {\text {wdim}}\;B \leq {\text {wdim}}\;H + {\text {wdim}}\;A and l.gldim B ≤ r.gldim H + l.gldim A {\text {l.gldim}} \; B \leq {\text {r.gldim}} \; H + {\text {l.gldim}} \; A .

Approximation of continuous vector functions

We study the possibility of uniform approximation of continuous mappings of metric compact sets into metric spaces. Notions of “weak dimension” and “weak Kolmogorov width” are introduced to compare approximating properties of infinite-dimensional subspaces. For classes of mappings specified by the majorant of the modulus of continuity, we present bilateral estimates of “weak” widths that may coincide under certain conditions.

On the exactness of flat resolvents

Let R be a left coherent ring, FP — idRR the FP — injective dimension of RR and wD(R) the weak global dimension of R. It is shown that 1) FP -idRR 0) if and only if every flat resolvent 0 → M → F° → F1... of a finitely presented right R—module M is exact at F'(i > n−1) if and only if every nth F -cosyzygy of a finitely presented right R — module has a flat preenvelope which is a monomorphism; 2) wD(R) 1) if and only if every (n−l)th F-cosyzygy of a finitely presented right R—module has a flat preenvelope which is an epimorphism; 3) wD(R) 0) if and only if every nth F — cosyzygy of a finitely presented right R — module is flat. In particular, left FC rings and left semihereditary rings are characterized

Categorical methods in graded ring theory

Let G be a group, R a G-graded ring and X a right G-set. We study functors between categories of modules graded by G-sets, continuing the work of [M]. As an application we obtain generalizations of Cohen-Montgomery Duality Theorems by categorical methods. Then we study when some functors introduced in [M] (which generalize some functors ocurring in [D1], [D2] and [NRV]) are separable. Finally we obtain an application to the study of the weak dimension of a group graded ring.

Global dimension of rings with krull dimension

The left global dimension of a semiprime ring, with left Krull dimension α≥1 is found to be the supremum of the projective dimensions of the p-critical cyclic modules where β≤α. A similar result is true for upper triangular matrix rings whose entries come from a domain with Krull dimension. In addition if R is a a ring of the form where S is a semiprime ring with left Krull dimension α≥1, T is any ring with l.K dim T≤α, and A is an S-T bimodule such that sA has Krull dimension then the left global dimension of R is the supremum of the projective dimensions of the -critical cyclic left R-modules where β<αa. These results are used to compute homological dimensions of rings with Krull dimension. Some analogues are given for weak dimension and for rings with Gabriel dimension.

Hereditary localizations of polynomial rings

and v= (fe R[x], c(f) = R)- = R[x] - U (mR[x]. 172 maximal ideal of R j. CT and f~’ are multiplicatively closed subsets of R[.x]. Set R(xj =R[s], and R(x) = R[x] v. Then R[x]c R(x) c R(s), R(s) is a localization of R( x >, and both R(x) and R(x) are faithfully flat R modules. Ever since R(x) played a prominent role in Quillen’s solution to Serre’s conjecture [25]? and its succeeding generalizations to non-Noetherian rings [6,20], there has been a considerable amount of interest in the properties of R<s). This interest expanded to include similarly constructed localizations of R[x]. Notable among these constructions is the ring R(x), which, through a variety of useful properties, provides a tool for proving results on R via passage to R(s). The interest in the properties of R(x) and R(x) branched in many directions [l-S, 7, 8: 12. 14-16, 21, 23, 26). Several of these directions consider homological properties of these two rings. Ferrand [IS] and McDonald and Waterhouse [23] investigate finitely generated projective modules over R(x). The behavior of the weak dimensions of R(x) and R(x) is investigated through the exploration of ascent and descent of Von Neumann regularity, semihereditarity and related conditions, and coherent regularity of the extensions R --f R(x) and R + R(x). In [II], Le Riche determines conditions for the semihereditarity of R(x). D. D. Anderson, D. F. Anderson, and Markanda 141, and Huckaba and Papick [15; $63 consider conditions related to semihereditarity such as being a Priifer or Priifcr-like ring, for both ring constructions. In Glaz 1121, we consider the

Weak dimension of group-graded rings

Weak dimension of group-graded rings