Stratiform Simple Artinian Rings
Stratiform Simple Artinian Rings
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层状简单海天环
DOI：
10.1112/plms/s3-53.2.267
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发表时间：
1986
影响因子：
1.8
通讯作者：
A. Schofield
中科院分区：
文献类型：
--
作者：
A. Schofield
关于某些Noether环如Weyl代数和多圈-有限群的群环的分式的斜域，已有许多论文，但很少有人对这种斜域提供某种相干理论;本文的目的是建立这样一个一般理论，并证明一些定理的推广，如Resco的嵌入定理[3]。和斯塔福德的定理[8]，证明了有限多圈群的Hirsch长度是其群环的分式斜域的不变量。我们将考虑一类斜域，即层状斜域，它包括有限多圈群环的分式的斜域和代数或可解李代数的包络代数的斜域，对于这些斜域，我们将证明一个与Resco的结果相对应的嵌入定理;此外，我们将发现，如果 * Dz> E是斜场的一个扩张，其中D或E是层状的，并且[D：E] t或[D：E] t中的一个维数是：E] r是有限的，那么另一个维度也是有限的。最后，在最后一节中，我们将发现，对于第一个Weyl斜域，任何包含中心的斜子域要么在中心上是超越度为1的交换的，要么使得Weyl斜域在它的两侧都是有限维的，通过本文，所考虑的所有环实际上都是对某个固定交换域k的fc-代数。此外，如果B是（R ′，R ′）-对于环R和R '的双模，我们还将假定k在左边的作用与k在右边的作用相同。这意味着（/？',/?)-双模自然等价于右R '® kR-模范畴，在这个等价下，秩为1的自由双模R' < 8>fc/？对应于秩为1的自由R '® kR-模。因此很清楚什么是投射双模或平坦双模。进一步，我们可以定义双模的同调维数和弱维数。我们常常把同调维数和弱维数分别表示为hd和wd。
There have been a number of papers on the skew fields of fractions of certain noetherian rings such as the Weyl algebras and the group rings of polycyclic-byfinite groups, but little has been done to provide some kind of coherent theory for such skew fields; it is the aim of this paper to set up such a general theory and to prove generalizations of a number of theorems such as Resco's embedding theorem [3] for purely transcendental commutative fields and Stafford's theorem [8] showing that the Hirsch length of a polycyclic-by-finite group is an invariant of the skew field of fractions of its group ring. We shall consider a class of skew fields, the stratiform skew fields, which include the skew fields of fractions of polycyclic-by-finite group rings and of enveloping algebras of algebraic or soluble Lie algebras; for these skew fields, we shall prove an embedding theorem corresponding to Resco's results; also, we shall find that if* D z> E is an extension of skew fields where either D or E is stratiform, and one of the dimensions [D: E] t or [D: E] r is finite, then the other dimension is also finite. Finally, in the last section, we shall find that for the first Weyl skew field, any skew subfield containing the centre is either commutative of transcendence degree 1 over the centre or else such that the Weyl skew field is finite-dimensional over it on both sides.Through this paper, all rings considered will actually be fc-algebras for some fixed commutative field k. Further, if B is an (R',/?)-bimodule for rings R and R', we shall also assume that the action of k on the left is the same as the action of k on the right. This means that the category of (/?',/?)-bimodules is naturally equivalent to the category of right R'® kR-modules and under this equivalence the free bimodule of rank 1, R'< 8> fc/? corresponds to the free R'® kR-module of rank 1. It is therefore clear what is meant by a projective bimodule or a flat bimodule. Further, we can define the homological dimension of bimodules and the weak dimension. We shall often abbreviate homological dimension and weak dimension to hd and wd respectively.