Vol. 30, No. 3, 1969

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A note on the theory of primes

Joel E. Schneider

Vol. 30 (1969), No. 3, 805–810

In this paper we find those commutative rings for which the theory of primes is subsumed under classical ideal theory, that is, for which every finite prime is an ideal. The characterization is given in terms of domains with this property and they are shown to form a class of domains from number theory. In addition we give two characterizations of the primes of a subring of a global field. The first establishes them as the nontrivial preprimes whose complements are multiplicatively closed and the second relates the space of all primes to that of the quotient field.

The concept of a prime for commutative rings with identity was introduced by Harrison in 1966.

Mathematical Subject Classification
Primary: 13.20
Received: 1 December 1968
Published: 1 September 1969
Joel E. Schneider