The direct summand conjecture for some bigenerated extensions and an asymptotic version of Koh’s conjecture

Abstract

This article deals with two different problems in commutative algebra. In the first part we give a proof of the direct summand conjecture 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”.