On isomorphic polynomial rings over graded rings
A graded ring R is called g-invariant if whenever a graded ring S is given and there exists a graded isomorphism of \(R[X_ 1,...,X_ n]\) onto \(S[Y_ 1,...,Y_ n]\) for some n, then R is isomorphic to S as a graded ring. A graded ring R is called strongly g-invariant if whenever a graded ring S and a graded isomorphism \(\phi\) of \(R[X_ 1,...,X_ n]\) onto \(S[Y_ 1,...,Y_ n]\) for some n are given, then \(\phi(R)=S\). The following holds: Let \(R=\oplus^{\infty}_{i=0}R_ i\) and \(S=\oplus^{\infty}_{i=0}S_ i\) be graded rings, where \(R_ 0\) and \(S_ 0\) are quasi-local rings. Let \(\phi\) be a graded isomorphism of \(R[X_ 1,...,X_ n]\) onto \(S[Y_ 1,...,Y_ n]\), where each \(X_ i\) and \(Y_ i\) are of weight 1. Then there is either an S-automorphism \(\psi\) of \(S[Y_ 1,...,Y_ n]\) of degree 0 satisfying \(\psi(Y_ i)=\phi(X_ i)\) for 1\(\leq i\leq n\), or R (resp. S) contains a graded subring A (resp. B) and a transcendental element x over A (resp. y over B) of weight 1 such that \(R=A[x]\) (resp. \(S=B[y])\). As corollaries it is proved that (1) if a graded ring \(R=\oplus^{\infty}_{i=0}R_ i\) is noetherian and \(R_ 0\) is a local ring, then R is g-invariant, and (2) if a graded ring \(R=\oplus^{\infty}_{i=0}R_ i\) is reduced and \(R_ 0\) is a quasi-local ring, then R is either strongly g-invariant or R is isomorphic to a polynomial ring over a graded ring in one variable.
cancellation for graded polynomial rings
scientific article; zbMATH DE number 3867472
scientific article