Bilinear maps on Artinian modules

TL;DR

This paper proves that nondegenerate bilinear maps between Artinian modules over a commutative ring imply the modules have finite length, and discusses limitations of extending this result to noncommutative cases.

Contribution

It establishes a new result linking nondegeneracy and finite length of Artinian modules over commutative rings, and explores the boundaries of this property in noncommutative contexts.

Findings

Nondegenerate bilinear maps imply finite length for Artinian modules over commutative rings.

Counterexamples show limitations in generalizing to noncommutative rings.

Raises open questions about possible extensions of the main theorem.

Abstract

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.