An affine version of a theorem of Nagata

TL;DR

This paper proves that certain affine rings can be embedded into polynomial rings, extending previous results and providing a new proof of the cancellation theorem for rings of transcendence degree one in characteristic zero.

Contribution

It introduces an affine version of Nagata's theorem, showing embeddings of rings into polynomial rings under specific conditions, and offers a concise proof of the cancellation theorem.

Findings

Rings with non-trivial embeddings into polynomial rings can be embedded into larger polynomial rings.

The theorem applies to subrings with non-zero locally nilpotent derivations.

Provides a new proof of the cancellation theorem for transcendence degree one rings.

Abstract

Let R be an affine k-domain over the field k. The paper's main result is that, if R admits a non-trivial embedding in a polynomial ring K[s] for some field K containing k, then R can be embedded in a polynomial ring F[t] which extends R algebraically. This theorem can be applied to subrings of a ring which admits a non-zero locally nilpotent derivation. In this way, we obtain a concise new proof of the cancellation theorem for rings of transcendence degree one for fields of characteristic zero.