Invariant dimension property of a ring $R$ which admits a homomorphism to a division ring $D$

Question

Let $R$ be a ring which admits a homomorphism to a division ring $D$. I know that if the homomorphism is surjective, then the ring has invariant dimension property. But if the homomorphism is not surjective, is it still true that $R^n$ isomorphic to $R^m$ as $R$-modules only when $n=m$? If this is not true, any counterexample? (The ring $R$ has identity.)

Answer 1

In fact, much more is true:

If $f:R\to S$ is a ring homomorphism, and $S$ has IBN property, then $R$ also has the IBN property.

This follows immediately form the following characterization of IBN rings: a ring $R$ has the IBN property iff for $A\in M_{m\times n}(R)$ and $B\in M_{n\times m}(R)$ with the property $AB=I_m$ and $BA=I_n$ it follows $m=n$. (See also Lam, Lectures on Modules and Rings, Remark 1.5.)

Answer 2

It is true if the homomorphism is injective, because:

• Stably finite rings (i.e. rings for which, for any square matrices $A$ and $B$, $AB=I\implies BA=I$) have the invariant basis number property,
• Division rings are stably finite,
• A subring of a stably finite ring is stably finite.