Completely Integrally Closed Domains
Let A be a domain and K its field of fractions. x in K is said to be almost integral over A if there is a such that for all . Then A is said to be completely integrally closed if every almost integral element of K is contained in A. A completely integrally closed domain is integrally closed. Conversely, a noetherian integrally closed domain is completely integrally closed.
Assume A is completely integrally closed. Then the formal power series ring is completely integrally closed. This is significant since the analog is false for an integrally closed domain: let R be a valuation domain of height at least 2 (which is integrally closed.) Then is not integrally closed. Let L be a field extension of K. Then the integral closure of A in L is completely integrally closed.
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