Integrally Closed Subrings of an Integral Domain

Let D be an integral domain with identity having quotient field K. This paper gives necessary and sufficient conditions on D in order that each integrally closed subring of O should belong to some subclass of the class of integrally closed domains ; some of the subclasses considered are the completely integrally closed domains, Prüfer domains, and Dedekind domains. 1. The class of integrally closed domains contains several classes of domains which are of fundamental importance in commutative algebra. Unique factorization domains, Krull domains, domains of finite character, Prüfer domains, completely integrally closed domains, Dedekind domains, and principal ideal domains are examples of such subclasses of the class of integrally closed domains. This paper considers the problems of determining, conversely, necessary and sufficient conditions on an integral domain with identity in order that each of its integrally closed subrings should belong to some subclass of the class of integrally closed domains. An example of a typical result might be Theorem 2.3: If J is an integral domain with identity having quotient field K, then conditions (1) and (2) are equivalent. (1) Each integrally closed subring of J is completely integrally closed. (2) Either J has characteristic 0 and K is algebraic over the field of rational numbers or J has characteristic p^O and K has transcendence degree at most one over its prime subfield. If J is integrally closed, then conditions (1) and (2) are equivalent to: (3) Each integrally closed subring of J with quotient field K is completely integrally closed. In considering characterizations of integral domains with identity for which every integrally closed subring is Dedekind or almost Dedekind (§3), we are led to use some results of W. Krull to prove Theorem 4.1, which establishes the existence of, as well as a method for constructing, a field with certain specified valuations. We then use this theorem to construct an example of an infinite separable algebraic extension field K of iIv(X) such that the integral closure J of fIp[A'] in K Received by the editors April 7, 1970. AMS 1969 subject classifications. Primary 1315, 1350; Secondary 1320.