Valuation Rings with Zero Divisors

Abstract

Manis has developed a valuation theory for commutative rings which extends valuation theory for fields. However his results do not extend the characterization of valuation rings as domains of maximal partial homomorphisms. In this note we show that Manis’ theory also generalizes this aspect of valuation theory. Next we show that an overring W of a valuation ring V is not necessarily a valuation ring in any nice sense, but that W is contained in a valuation ring which is a large quotient ring of V.