Vol. 54, No. 1, 1974

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A Nullstellensatz for Nash rings

Gustave Adam Efroymson

Vol. 54 (1974), No. 1, 101–112

Let D be a domain in Rn defined by a finite number of strict polynomial inequalities. Then the Nash ring AD is the ring of real valued algebraic analytic functions defined on D. In this paper, it is shown that AD is Noetherian and has a nullstellensatz. For 𝒫 a prime ideal of AD,AD𝒫 is said to be rank one orderable if its quotient field can be ordered over R so that it has essentially one infinitesimal. Then AD𝒫 is rank one orderable if and only if 𝒫 equals the set of functions in AD which vanish on the zero set of 𝒫 in D.

Mathematical Subject Classification 2000
Primary: 14A25
Secondary: 32C05
Received: 16 April 1973
Revised: 18 September 1973
Published: 1 September 1974
Gustave Adam Efroymson