### Regular Overrings of Regular Local Rings

Transactions of the American Mathematical Society

Vol. 171, September 1972, pp291-300.

Abstract

The local factorization theorem of Zariski and Abhyankar characterizes all 2-dimensional regular local rings which lie between a given 2-dimensional regular local ring *R* and its quotient field as finite quadratic transforms of *R*. This paper shows that every regular local ring *R* of dimension n > 2 has infinitely many minimal regular local overrings which cannot be obtained by a monoidal transform of *R*. These overrings are localizations of rings generated over *R* by certain quotients of elements of an *R*-sequence. Necessary and sufficient conditions are given for this type of extension of *R* to be regular.

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