# Answer to Question #24892 in Abstract Algebra for Melvin Henriksen

Question #24892

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.

Expert's answer

Assuming

finite product of prime ideals is nonempty. Let

*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 afinite 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*,. . . ,*p*n*such that p1*· · ·*p*n*= 0. We claim thatany minimal prime p is among the p*i*’s. Indeed, from p1*· · ·*p*n**⊆**p, we must have p**i**⊆**p for some**i*. Hence p = p*i*.Therefore,*R*has at most*n*minimal primes.
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