TITLE: Help with a problem about Artinian rings from Christian Peskine's book
QUESTION [9 upvotes]: I am stuck with problem 3, chapter 4, from the book of Peskine, An Algebraic Introduction to Complex Projective Geometry.

Let $A$ be a Noetherian ring. Assume that if $a \in A$ is neither invertible nor nilpotent, then there exist $b \in A$ such that $b$ is not nilpotent and that $ab = 0$. Show that $A$ is Artinian.

I have tried to prove that every prime is also maximal since I already have the Noetherian property and I can't see how to prove directly the dcc.

REPLY [5 votes]: Here is my answer. Assume that $A$ is not artinian. Then there exists a maximal ideal not contained in the union of all minimal primes. Hence
there exists an element $a$ not invertible and not contained in any minimal prime.
Consequently if $ab=0$ then $b$ is contained in all minimal primes and is therefore nilpotent, contradicting the hypothesis.