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A Characterization of Finite Commutative Roger Alperin and Eloise Hamann

Summary: A Characterization of Finite Commutative
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


Source: Alperin, Roger C. - Department of Mathematics, San Jose State University


Collections: Mathematics