Tensor products of faithful modules

TL;DR

The paper investigates conditions under which the tensor product of faithful modules over algebras remains faithful, extending known results from fields to general commutative rings and exploring algebra-structure-free versions.

Contribution

It generalizes the faithfulness of tensor products from fields to commutative rings and introduces a version avoiding algebra structures on the modules.

Findings

Tensor product of faithful modules over fields is faithful.

Certain conditions on algebras and modules ensure faithfulness over rings.

A version of the main result does not require algebra structures.

Abstract

If $k$ is a field, $A$ and $B$ $k$-algebras, $M$ a faithful left $A$-module, and $N$ a faithful left $B$-module, we recall the proof that the left $A\otimes_k B$-module $M\otimes_k N$ is again faithful. If $k$ is a general commutative ring, we note some conditions on $A,$ $B,$ $M$ and $N$ that do, and others that do not, imply the same conclusion. Finally, we note a version of the main result that does not involve any algebra structures on $A$ and $B.$