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# Answer to Question #24891 in Abstract Algebra for Melvin Henriksen

Question #24891
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.
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&rsquo;s Lemmato the family of prime ideals &sube; p. It suffices to check that, forany chain of prime ideals {pi : i &isin; I} in p, their intersection p&#039; is prime.Let a, b not in p&#039;. Then a not in pi and
b not in pj for some i, j &isin; I. If, say, pi &sube; pj , then both a, b are outside pi, so we have arbnot in pi for some r &isin; R. But then arb not in p&#039;, and we have checked thatp&#039; is prime.

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