Weak Global Dimension Research Articles

Coherent rings of finite weak global dimension

Coherent rings of finite weak global dimension

Arithmetical and semihereditary semigroup rings

In this paper conditions on the commutative ring R (with identity) and the commutative semigroup ring S (with identity) are found which characterize those semigroup rings R[S] which are reduced or have weak global dimension at most one. Likewise, those semigroup rings R[S] which are semihereditary are completely determined in terms of R and S.

On the Weak Global Dimension of Pseudovaluation Domains

In [7], Hedstrom and Houston introduce a type of quasilocal integral domain, therein dubbed a pseudo-valuation domain (for short, a PVD), which possesses many of the ideal-theoretic properties of valuation domains. For the reader′s convenience and reference purposes, Proposition 2.1 lists some of the ideal-theoretic characterizations of PVD′s given in [7]. As the terminology suggests, any valuation domain is a PVD. Since valuation domains may be characterized as the quasilocal domains of weak global dimension at most 1, a homological study of PVD's seems appropriate. This note initiates such a study by establishing (see Theorem 2.3) that the only possible weak global dimensions of a PVD are 0, 1, 2 and ∞. One upshot (Corollary 3.4) is that a coherent PVD cannot have weak global dimension 2: hence, none of the domains of weak global dimension 2 which appear in [10, Section 5.5] can be a PVD.

Flache darstellungen von geordneten mengen

We consider categories [I,A] of diagrams over an ordered set I with values in a module category A. Using a description of the flat diagrams over downward directed trees we characterize trees to be exactly the ordered sets I with the property that the difference of weak global dimensions w.gl.dim[I,A] — w.gl.dim A is bounded by 1. In this case we also compute upper bounds for the global dimension of [I,A].

Coherence and weak global dimension of R[[X]] when R is von Neumann regular

dimension of R. Very little is known about what can occur except that the weak global dimension must increase by at least one and does increase by exactly one when R[[X]] is coherent. This paper attempts to cast some light on this problem and the closely related question of the stability of coherence under the formation of the power series ring. It is devoted to determining necessary and sufficient conditions on a commutative (von Neumann) regular ring R for the power series ring R[[X]] to b e coherent (equivalently, semihereditary) and also conditions for R[[X]] t o h ave weak global dimension one. Surprisingly, it turns out that R[[X]] h as weak global dimension one precisely when R[[X’j] is a B&out ring so that property is characterized as well. Each of these properties is characterized in several ways both in terms of internal conditions on R and conditions that involve the category of R-modules. Perhaps the conditions which can be most readily verified for a specific ring are expressed in terms of a natural partial order z< which is defined on R by a < b if and only if ab = a2 for a, b in R. For example, it is shown that R[[X]] is coherent if and only if every countable subset of R that forms a chain in the partial order < on R has a least upper bound in R. As further illustrations of our results we mention that R[[Xj] h as weak global dimension one precisely in case R satisfies either of two limited forms of self-injectivity and also in case R satisfies a restricted type of algebraic compactness. These results permit us to give an example that shows R[[X]] can have weak global dimension one without being coherent even when R is a Boolean ring. This settles a question raised by Jensen [7]. Another example is included to illustrate the added complexity that occurs in arbitrary regular rings as compared with Boolean rings. This example also provides a pair of regular rings R contained in S, and hence a faithfully flat extension, such that R[[X]] is not pure in S[[X]]. Thus it gives a negative

Separable algebras over Prufer domains

Employing sheaf-theoretic techniques and idempotent lifting properties, the following extension of the Wedderburn Principal Theorem is shown: If A is any finitely generated algebra over an almost Dedekind domain R such that A modulo its Baer lower radical is separable over R, then A contains a separable subalgebra S which adds with the lower radical as R-modules to give A. Certain structural results concerning separable algebras over arbitrary Prufer domains are obtained. These parallel results already known for Dedekind domains. Finally, results relating the Hochschild and the weak global dimensions of an algebra with the weak global dimension of the ground ring are obtained. The ground rings include some Prufer domains and their generalizations. The purpose of this article is to examine the structure of commutative and non-commutative algebras over Prufer domains, and in particular over almost Dedekind domains. We shall prove for these last rings a generalization of the Wedderburn Principal Theorem. In § 1, it will be shown that every finitely generated, torsion-free, separable commutative algebra S over a Prufer domain R is again a Prufer domain. If R is an almost Dedekind domain, then so also is S. In § 2, similar results will be obtained for non-commutative separable algebras over these same types of ground ring. Moreover, since a finitely generated, torsion-free algebra over a Prufer domain is projective, one may write any separable algebra over a Prufer domain as a direct sum of the torsion ideal and the projective ideal. This decomposition can be lifted to any finitely generated algebra A which is separable modulo its lower radical. An application of sheaf-theoretic techniques as in [3] to this decomposition obtains the Wedderburnlike structure theorem for almost Dedekind domains. In § 3, we will give results relating the Hochschild dimension and the weak global dimension of a finitely generated, torsion-free algebra Received by the editors August 9,1973. AMS Subject Classification (1970): Primary 13F05, 16A16, 16A46; Secondary 13E10, 16A32, 16A60.

Algebras Over Absolutely Flat Commutative Rings

Let A be a finitely generated algebra over an absolutely flat commutative ring. Using sheaf-theoretic techniques, it is shown that the weak Hochschild dimension of A is equal to the supremum of the Hochschild dimension of ${A_x}$ for x in the decomposition space of R. Using this fact, relations are obtained among the weak Hochschild dimension of A and the weak global dimensions of A and ${A^e}$. It is also shown that a central separable algebra is a biregular ring which is finitely generated over its center. A result of S. Eilenberg concerning the separability of A modulo its Jacobson radical is extended. Finally, it is shown that every homomorphic image of an algebra of weak Hochschild dimension 1 is a type of triangular matrix algebra.

Algebras over absolutely flat commutative rings

Let A be a finitely generated algebra over an absolutely flat commutative ring. Using sheaf-theoretic techniques, it is shown that the weak Hochschild dimension of A is equal to the supremum of the Hochschild dimension of A x {A_x} for x in the decomposition space of R. Using this fact, relations are obtained among the weak Hochschild dimension of A and the weak global dimensions of A and A e {A^e} . It is also shown that a central separable algebra is a biregular ring which is finitely generated over its center. A result of S. Eilenberg concerning the separability of A modulo its Jacobson radical is extended. Finally, it is shown that every homomorphic image of an algebra of weak Hochschild dimension 1 is a type of triangular matrix algebra.

A commutative local ring with finite global dimension and zero divisors

It is well known that a commutative, local, noetherian ring (with 1) of finite global dimension must be a domain (indeed a regular local ring). For commutative local rings which are coherent or have linearly ordered ideals, finite global dimension also implies no proper zero divisors (see [6] and [7]). In this note we show that these results cannot be generalized to arbitrary commutative local rings. An example is given of a ring with global dimension 3 possessing a localization which is not a domain. This ring also has weak global dimension 2, these dimensions being the smallest possible in local rings with zero divisors. Some results on homological dimension of ideals generated by two elements in C(R), the ring of all continuous real-valued functions on the reals R, are also included.

The Weak Global Dimension of the Group-Rings of Abelian Groups

Journal of the London Mathematical SocietyVolume s1-36, Issue 1 p. 371-381 Notes and papers The Weak Global Dimension of the Group-Rings of Abelian Groups A. J. Douglas, A. J. Douglas The University SheffieldSearch for more papers by this author A. J. Douglas, A. J. Douglas The University SheffieldSearch for more papers by this author First published: 1961 https://doi.org/10.1112/jlms/s1-36.1.371Citations: 5AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volumes1-36, Issue11961Pages 371-381 RelatedInformation

On regular group rings

Let G be a multiplicative group, K a commutative ring with unit, and K(G) the group ring of G with respect to K. We say that K(G) is regular if given an x in K(G), there is a y in K(G) such that xyx = x. Using a homological characterization of regular rings which was found independently by M. Harada [2, Theorem 5] and the author, we prove that if G is locally finite, then K(G) is regular if and only if K is regular and is uniquely divisible by the order of each element in G. More generally we show that if K(G) is regular, then G is a torsion group and K is a regular ring which is uniquely divisible by the order of each element in G. A nonhomological proof of these results has been given by J. McLaughlin (unpublished). In conclusion, we show that if G is a commutative group and K is a field of characteristic not dividing the order of any element in G, then the weak global dimension of K(G) equals the rank of G. For the most part we follow the conventions and notations in [']. Let R be a ring (with unit) and A a left R-module. The weak left dimension of A is defined as follows (see [1, Chapter VI, Exercise 3]):