A note on finite dimensional subrings of polynomial rings
A note on finite dimensional subrings of polynomial rings
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关于多项式环的有限维子环的注解
DOI：
10.1090/s0002-9939-1972-0289498-2
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发表时间：
1972
期刊：
影响因子：
--
通讯作者：
P. Eakin
中科院分区：
文献类型：
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作者：
P. Eakin
设k是域，{Xn}是k上的有限族.证明了若A是Krull维数为d的环，使得kcAck [{A0 ahA]，则存在在k上代数独立的元素Y，* Yd和k-同构q，使得kc+（A）ck [Y1，**，Yd].这是用来表明，一个一维环A，满足上述条件的必然是一个仿射环在k和必然是多项式环，如果它是正常的。此外，我们还证明了这样的环A是k上超越度为2的正规仿射环当且仅当它是二维Krull环，使得A的每个本质赋值都有k上超越的剩余域.导论.在[EZ] Evyatar和Zaks问是否Dedekind域之间的字段k和多项式在有限数量的变量在k必然是一个多项式环。在这里，我们表明，答案是肯定的，没有限制的变量的数量，并没有假设环满足所有三个诺特公理上的戴德金域（它只需要是一维和整体封闭）。此外，我们还给出了域上多项式的子环与Noether环之间的关系的一些一般性结果。作者感谢W。J. Heinzer的利益，许多刺激性的谈话就这个问题和肯尼斯久保田的建议，简化了证明引理B。1.在下面的所有内容中，一个环将总是被假定为交换的，并拥有一个乘法单位元1 # 0。对于维数A，我们总是指环A的Krull维数。引理A（化简为有限个变量的情形）。设D是整环，{Xa}aEA是D上的不定式族.设A是环，使得（1）D？A < D[{Xa}aEA. (2)A满足d.c.c.关于prime ideals 1971年3月3日由编辑接收。AMS 1969主题分类。小学1635，1230，1615。
Let k be a field and {Xn}aeA a familyofindeterminates over k. We show that if A is a ring of Krull dimension d such that k c A c k[{A0ahA] then there are elements Y,* Yd which are algebraically independent over k and a k-isomorphism q such that k c +(A) c k[Y1, **, Yd]. This is used to show that a onedimensional ring A which satisfies the above conditions is necessarily an affine ring over k and is necessarily a polynomial ring if it is normal. In addition we show that such a ring A is a normal affine ring of transcendence degree two over k if and only if it is a two-dimensional Krull ring such that each essential valuation of A has residue field transcendental over k. Introduction. In [EZ] Evyatar and Zaks ask whether a Dedekind domain which is caught between a field k and the polynomials in a finite number of variables over k is necessarily a polynomial ring. Here we show that the answer is affirmative with no restriction on the number of variables and without assuming the ring satisfies all three of the Noether axioms on a Dedekind domain (it need only be one dimensional and integrally closed). In addition we give some general results which relate subrings of the polynomials over a field to noetherian rings. The author is indebted to W. J. Heinzer for the benefit of many stimulating conversations on this subject and to Kenneth Kubota for a suggestion which simplified the proof of Lemma B. 1. In all that follows, a ring will always be assumed to be commutative and to possess a multiplicative identity 1 # 0. By dimension A we always refer to the Krull dimension of the ring A. LEMMA A (Reduction to the case of a finite number of variables). Let D be an integral domain and {Xa}aEA a family of indeterminates over D. Let A be a ring such that (1) D ? A < D[{Xa}aEA. (2) A satisfies the d.c.c. on prime ideals. Received by the editors March 3, 1971. AMS 1969 subject classifications. Primary 1635, 1230, 1615.