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This article deals with two different problems in commutative algebra. In thefirst 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 UniqueFactorization Domain of mixed characteristic zero. We assume that $S$ isgenerated by two elements which satisfy, either radical quadratic equations, orgeneral quadratic equations under certain arithmetical restrictions. In thesecond part of this article, we discuss an asymptotic version of Koh'sConjecture. We give a model theoretical proof using "non-standard methods".

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