Generalizations of the Direct Summand Theorem over UFD-s for some Bigenerated Extensions and an Asymptotic Version of Koh's Conjecture

TL;DR

This paper proves generalized forms of the Direct Summand Theorem for certain module-finite extensions over UFDs of mixed characteristic and discusses an asymptotic version of Koh's Conjecture using non-standard methods.

Contribution

It introduces new generalized forms of the Direct Summand Theorem for specific extensions and provides an asymptotic approach to Koh's Conjecture with model-theoretic proofs.

Findings

Generalized Direct Summand Theorem for extensions with quadratic relations

Asymptotic version of Koh's Conjecture proved using non-standard methods

Extension rings over UFDs of mixed characteristic satisfy new splitting properties

Abstract

This article deals with two different problems in commutative algebra. In the first part, we give a proof of generalized forms of the Direct Summand Theorem (DST (or DCS)) for module-finite extension rings of mixed characteristic $R\subset S$ satisfying the following hypotheses: The base ring $R$ is a Unique Factorization Domain of mixed characteristic zero. We assume that $S$ is generated by two elements which satisfy, either radical quadratic equations, or general quadratic equations under certain arithmetical restrictions. In the second part of this article, we discuss an asymptotic version of Koh's Conjecture. We give a model theoretical proof using "non-standard methods".