A Characterization of Finite Commutative Roger Alperin and Eloise Hamann Summary: A Characterization of Finite Commutative Rings Roger Alperin and Eloise Hamann We give a partial converse to the well-known result: `If R is finite commuta- tive ring with identity then every element is a unit or a zero divisor'. An important partial converse of this which we use here is that: `A field which is finitely gen- erated as a ring is finite'. We hope students of commutative algebra may find our proof interesting and enlightening. Theorem 1 Let R be a commutative ring with unity which is finitely generated as a ring. If evey element of R is either a unit or a zero divisor then R is finite. Proof: Let Z(R) be the zero divisors of the ring R. Since a finitely generated ring is a quotient of a polynomial ring over Z in finitely many indeterminates, it is Noetherian. The primary decomposition theorem gives (0) = n k=1 Ik for primary ideals Ik; rad(Ik) = Pk. As a consequence Z(R) = n i=1 Pi, [1, Prop. 4.7]. It follows immediately from the hypothesis that any proper ideal I of R can consist only of zero divisiors. Thus I n i=1 Pi; so using [1, Prop. 1.11] I Pi for some i. In particular all maximal ideals are among the Pi, 1 i n. Hence there Collections: Mathematics