Content Algebras
Content Algebras
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发表时间：
1978
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通讯作者：
David E. Rush
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作者：
David E. Rush
设R是有单位元的交换环，X是不定环。文[5]和[16]证明了如果f，g∈R[X]，则对某个整数n≥L，c(F)n+1c(G)=c(G)NC(Fg)，其中c(H)表示由h∈R[X]的系数生成的JR的加子群。实际上，[5J]和[16]中的陈述并不像这样笼统；然而，证明是这样的。具体地说，Merten在[16]中只考虑了R是整数上的多项式环的情况，但这通过专门化f和g的系数给出了任何环的结果。在[5]中Dedekin只考虑了JR是代数整数环的情况，但他的证明是完全通用的。此外，如果设c(H)表示由h∈R[X]的系数生成的R的S子模，则上述公式成立，S是R的一个子环，它通常以这种形式出现，特别是在S=R的情况下。Dedekind非常优雅的证明在[15，第9页，引理6.1]中得到了重现。
Let R be a commutative ring with identity and let X be an indeterminate. In [5] and [16] it was shown that if f, g∈R[X], then for some integer n≥l, c(f)n+1c(g) = c(g)nc(fg), where c(h) denotes the additive subgroup of JR generated by the coefficients of h ∈ R[X]. Actually the statements in [5J and [16] are not so general as this; however, the proofs are. Specifically, in [16] Mertens considers only the case that R is a polynomial ring over the integers, but this gives the result for any ring by specializing the coefficients of f and g. In [5] Dedekind considers only the case that JR is a ring of algebraic integers, but his proof is completely general. Further, the above formula then holds if one lets c(h) denote the S-submodule of R generated by the coefficients of h ∈ R[X], S a subring of R, and it is in this form that it usually appears, especially the case S = R. Dedekind′s very elegant proof is reproduced in [15, p. 9, Lemma 6.1].