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.
