A property of the differential ideal
A property of the differential ideal
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微分理想的性质
DOI：
10.1090/s0002-9947-1960-0113880-3
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发表时间：
1960
影响因子：
1.3
通讯作者：
K. B. O’Keefe
中科院分区：
文献类型：
--
作者：
K. B. O’Keefe
导言。设y是有理数域R上的不定微分，即考虑(代数)不定序列中的多项式环R[Yo，Y1，Y2，*]，以及R[Y0，Y1，Y2，***i到自身的映射a--a‘，它具有如下性质：(1)(a+b)’=a‘+b’，(2)(Ab)‘=a’b+ab‘，(3)Yj=yi+i；这样的映射只有一个，从a传递到a‘的操作称为区分。R[YO，Y1，Y2]中的微分理想指的是环论意义上的理想，它具有这样的性质：如果a在理想中，则a‘也在理想中。通常，[yp]代表由yp生成的微分理想，即对于由yp生成的通常环论意义上的理想，(Yp)，(Yp)，*即使对于相对简单的理想[yp]，研究微分理想的结构也会产生许多悬而未决的问题。简单的计算表明：y2-O=0[yp]，由此推论每个yi的某一次幂在[yp]中。J·F·里特[3]挑出了以下问题进行研究：最小的Q是什么，使得y=O[Yp]？对于i=1，q=2p-1是他在没有证据的情况下所说的答案。在第一部分，我们给出了这一结果的一个证明，在第二部分，我们解决了i=2，p>；2的问题。对于任意的i，我们猜想它的答案是q=(i+1)(p-1)+1。让P o yy‘.。。Y‘n是一个强大的产品(pp.)学位
Introduction. Let y be a differential indeterminate over the rational number field R, that is, we consider the polynomial ring R [yO, yl, Y2, * ] in a sequence of (algebraic) indeterminates yo =y, Yl, * * * together with the mapping a--a' of R [yO, Yl, Y2, * * * I into itself which has the properties: (1) (a +b)' = a'+b', (2) (ab)' = a'b +ab', (3) yJ = yi+i; there is one and only one such mapping, and the operation of passing from a to a' is called disfferentiation. By a differential ideal in R [yo, Yl, Y2, ] we mean an ideal in the ringtheoretic sense which has the property that if a is in the ideal, then also a' is in the ideal. Notationally, [yP] stands for the differential ideal generated by yP, that is, for the ideal generated in the usual ring-theoretic sense by yP, (yP), ((Yp)'), * A study of the structure of differential ideals yields many unsolved problems even for the relatively simple ideal [yP]. It is shown from a simple calculation that y2 -O=0[yP], whence it follows that some power of each yi is in [yP]. The following question was singled out for investigation by J. F. Ritt [3 ]: what is the smallest q such that y = O[yP]? For i= 1, q=2p-1 is stated by him without proof to be the answer. In Part I we give a proof of this result, and in Part II we solve the problem for i =2, p > 2. For arbitrary i we conjecture the answer to be q=(i+1)(p-1)+1. The following notation and results of H. Levi we use extensively. Let P o yy' . . . y'n be a power product (pp.) of degree