AF-整环及其推广
AF-domains and their generalizations
摘要：
本文研究域$k$上代数张量积的维数论。本文引入并研究了广义AF-Domain（简称GAF-Domain）的概念，证明了任何$k$ -代数$A$使得单变量多项式环$A[X]$是AF-Domain，特别是任何AF-Domain都是GAF-Domain.此外，我们还计算了$A\otimes_kB$的克鲁尔维数，对于任何$k$ -代数$A$，使得$A[X]$是AF-整环，对于任何$k$ -代数$B$，推广了Wadsworth在[16]中的主要定理.
英文摘要：
In this paper, we are concerned with the study of the dimension theory of tensor products of algebras over a field $k$. We introduce and investigate the notion of generalized AF-domain (GAF-domain for short) and prove that any $k$-algebra $A$ such that the polynomial ring in one variable $A[X]$ is an AF-domain is in fact a GAF-domain, in particular any AF-domain is a GAF-domain. Moreover, we compute the Krull dimension of $A\otimes_kB$ for any $k$-algebra $A$ such that $A[X]$ is an AF-domain and any $k$-algebra $B$ generalizing the main theorem of Wadsworth in [16].
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