Question

Show that any prime ideal p in a ring R contains a minimal prime ideal. Using this, show that the lower nilradical Nil*R is the intersection of all the minimal prime ideals of R.

Answer 1

The second conclusion followsdirectly from the first, since Nil*R is the intersection of all theprime ideals of R. To prove the first conclusion, we apply Zorn’s Lemmato the family of prime ideals ⊆ p. It suffices to check that, forany chain of prime ideals {pi : i ∈ I} in p, their intersection p' is prime.Let a, b not in p'. Then a not in pi and
b not in pj for some i, j ∈ I. If, say, pi ⊆ pj , then both a, b are outside pi, so we have arbnot in pi for some r ∈ R. But then arb not in p', and we have checked thatp' is prime.