Contracted ideals from integral extensions of regular rings

Introduction. The purpose of this paper is to consider the following question: if R is a regular Noetherian ring and S ⊃ R is a module-finite R -algebra, is R a direct summand of S as R -modules? An affirmative answer is given if R contains a field, and it is shown that if the completions of the local rings of S possess maximal Cohen-Macaulay modules in the sense of § 1 of 6 then the conclusion is valid in this case too. Hence, if Conjecture E of 6 is true then the question raised here has an affirmative answer without further restriction on the regular Noetherian ring R , and it will be shown here that only a greatly weakened version of Conjecture E is needed.