# 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.

Expert's answer

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**{*p*i*:*i**∈**I}*in p, their intersection p'*is prime.Let**a, b*not in*p**'*. Then*a*not in*p**i*and*b*not in*p**j*for some*i, j**∈**I*. If, say, p*i**⊆**p**j*, then both*a, b*are outside p*i*, so we have*arb*not in*p**i*for some*r**∈**R*. But then*arb*not in*p**'*, and we have checked thatp*'*is prime.
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