It is shown that if a bilinear map f: A x B --> C of modules over a
commutative ring k is nondegenerate (i.e., if no nonzero element of A
annihilates all of B, and vice versa), and A and B are Artinian, then A and B
are of finite length.

Some consequences are noted. Counterexamples are given to some attempts to
generalize the above statement to balanced bilinear maps of bimodules over
noncommutative rings, while the question is raised whether other such
generalizations are true.