摘要:

Let A be a commutative algebra over a field k, and VAbe the k-subalgebra of Endk(A) generated by EndA(A) = A and all k-derivations of A. A study of the homological properties of VAwas initiated by Hochschild, Kostant, and Rosenberg in [5], and continued by Rinehart [8], [9], Roos [11], Björk [1], Rinehart and Rosenberg [10], and others. It was proved in [5] that, if k is perfect and A is a regular affine algebra of dimension r, then the global dimension of VAis between r and 2r. Moreover, if k has positive characteristic, then gl.dim VA= 2r [8]. By a recent celebrated theorem of Roos [11], gl.dim VA= r if k has characteristic zero and A = k[x1, …, xr]; in this case VAis the so-called “Weyl algebra on 2r variables”.

ISSN:0092-7872    

DOI:10.1080/00927877408548623

出版商:Taylor & Francis Group    

年代:1974

数据来源: Taylor

摘要:

Simple locally compact rings without open left ideals were considered in [13] and general locally compact rings without open left ideals were studied extensively in [5] and [6]. We remove the hypothesis of local compactness and consider topological rings A without open left ideals but containing an open subring R. In section 4 we show that under these conditions A is completely determined by R. More precisely A can be identified with the topological ring of quotients C(R) introduced in [8]. As an R-moduleRA is topologically isomorphic to I*(RR), the topological injective hull ofRR. The last statement was proved in [6] for A locally compact and R compact. Section 5 gives a characterization of those linearly topologized rings R that can be openly embedded into a ring A without open left ideals. In particular we shall show that the open left ideals form an idempotent ideal filter with quotient ring A. In section 6 we consider the class ℋ of all topological rings that can be openly embedded into a topological ring without open left ideals. If we restrict our attention to linearly topologized rings, then ℋ is Morita-invariant. In section 2 we construct a topological ring of quotients Q*(R) and prove that it coincides with the ring C(R) of [8].

ISSN:0092-7872    

DOI:10.1080/00927877408548624

出版商:Taylor & Francis Group    

年代:1974

数据来源: Taylor

摘要:

In this paper, we study relations in general categories. Our approach requires that these must have finite products and factorization systems. Klein [5] has obtained a condition for composition of relations to be strictly associative. Here we consider the possibility that associativity only holds up to a coherent isomorphism, in other words that the relations are the morphisms of a bicategory in the sense of Bénabou [2].

ISSN:0092-7872    

DOI:10.1080/00927877408548625

出版商:Taylor & Francis Group    

年代:1974

数据来源: Taylor

摘要:

The purpose of this article is to determine the injective objects in some complete categories of rings. All rings are assumed to have identities and it is assumed that the homomorphisms preserve these identities. We recall that an object Q in a category is called injective if for every diagram where A′ → A is a monomorphism, there is a map A → Q making the triangle commute. The zero ring belongs to all the categories discussed and it is easy to see that it is an injective object. For the categories of commutative rings, strongly regular and commutative regular rings we show that the zero ring is the only injective by using the fact that an injective object must be a retract of any extension. We include in this section the known results which characterize the injective rings and p-rings. The second part of the paper discusses injectivity with respect to regular monomorphisms. Some necessary categorical background is given and it is then shown that results analagous with those of the first section hold (including the known Boolean and p-ring cases). In an abelian category all monomorphisms are regular, so in the study of the injective objects, for example injective modules, there are not two separate cases.

ISSN:0092-7872    

DOI:10.1080/00927877408548626

出版商:Taylor & Francis Group    

年代:1974

数据来源: Taylor

摘要:

Let A be a local ring of dimension d. If A is a quotient of a regular local ring of dimension n = d+r, then we say that A hasembeddingcodimension≤ r. This paper investigates some special properties of local rings of small embedding codimension. The main idea is to exploit a result in [17], which says that local rings of small embedding codimension and depth ≥ 3 are parafactorial. This tells us, with suitable additional hypotheses, that the ring is factorial, or Gorenstein, or even a complete intersection.

ISSN:0092-7872    

DOI:10.1080/00927877408548627

出版商:Taylor & Francis Group    

年代:1974

数据来源: Taylor