Answer to Question #24892 in Abstract Algebra for Melvin Henriksen

Question

Show that if the ideals in R satisfy ACC (e.g. when R is left noetherian), then R has only finitely many minimal prime ideals.

Answer 1

Assuming R <> 0, wefirst prove that any ideal in R contains a finite product of primeideals. Suppose, instead, that the family F of ideals which do not contain a
finite product of prime ideals is nonempty. Let I be a maximal member ofF. Certainly I <> R, and I is not prime. Therefore,there exist ideals A,B contains I such that AB ⊆ I. But each of A,B contains a finiteproduct of primes, and hence so does I, a contradiction. Applying theabove conclusion to the zero ideal, we see that there exist prime ideals p1,. . . , pn such that p1 · · · pn = 0. We claim thatany minimal prime p is among the pi’s. Indeed, from p1 · · · pn⊆ p, we must have pi ⊆ p for some i. Hence p = pi.Therefore, R has at most n minimal primes.