A note on the Hilbert basis theorem (Q1092983)

Let \(R=\oplus \sum _{i\in Z}R_ i\) be a Z-graded ring, not necessarily with identity; where Z is the additive group of integers. Let \(M=\oplus \sum _{i\in Z}M_ i\) be a Z-graded right R-module. The following are proved to be equivalent: (I) \(M_ R\) (forgetting the grading) is noetherian. (II) Each \(M_ i\) is a noetherian \(R_ 0\)-module and given any sequence \(\{N_ i\}_{i\in Z}\), where each \(N_ i\) is a subgroup of \(M_ i\) such that \(N_ iR_ j\subset N_{i+j}\), there exists a finite non-empty set F of integers such that for \(k\not\in F\), \(N_ k=\sum _{k\in F}N_ iR_{k-i}\). (III) Each \(M_ i\) is a noetherian \(R_ 0\)-module and the following hold. (a) Given any sequence \(\{N_ i\}_{i\geq 0}\), where each \(N_ i\) is a subgroup of \(M_ i\) and \(N_ iR_ j\subset N_{i+j}\) for \(j\geq 0\), there exists \(m\geq 0\) such that for \(k>m\), \(N_ k=\sum ^{m}_{i=0}N_ iR_{k-i}\) and (b) given any sequence \(\{N_ i\}_{i\leq 0}\), where each \(N_ i\) is a subgroup of \(M_ i\) and \(N_ iR_ j\subset N_{i+j}\) for \(j\leq 0\), there exists \(s\leq 0\) such that for \(k<s\), \(N_ k=\sum ^{0}_{i=s}N_ iR_{k- i}\). This result generalizes the Hilbert basis theorem. Let now M be positively graded in the sense that \(M_ i=0\) for \(i<0\) and also let R be positively graded. R gives rise to a ring \^R whose members are the sequences \(\{x_ i\}_{i\geq 0}\), \(x_ i\in R_ i\) and in which the addition and multiplication are the natural extensions of the corresponding operations in R. We further get a corresponding \^R-module \^M from M. It is proved that \(M_ R\) is noetherian if and only if \^M\({}_{\hat R}\) is noetherian. Some applications of the above results are given. One of them is that given any ring R, R[x] is right noetherian if and only if R is right noetherian and has right identity.