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.
Read more about this topic: Integrally Closed Domain
Famous quotes containing the words completely, closed and/or domains:
“The atmosphere of orthodoxy is always damaging to prose, and above all it is completely ruinous to the novel, the most anarchical of all forms of literature.”
—George Orwell (1903–1950)
“Had I made capital on my prettiness, I should have closed the doors of public employment to women for many a year, by the very means which now makes them weak, underpaid competitors in the great workshop of the world.”
—Jane Grey Swisshelm (1815–1884)
“I shall be a benefactor if I conquer some realms from the night, if I report to the gazettes anything transpiring about us at that season worthy of their attention,—if I can show men that there is some beauty awake while they are asleep,—if I add to the domains of poetry.”
—Henry David Thoreau (1817–1862)