This is a follow-up to this question of mine: Cardinality of a minimal generating set is the cardinality of a basis
I observed in a comment to that question that it was, in fact, a duplicate of this question: Given a commutative ring $R$ and an epimorphism $R^m \to R^n$ is then $m \geq n$? (1)
The proof uses heavily that the ring is commutative.
I am wondering now of an example of a ring with IBN where the answer to (1) is negative.
Arturo's answer to my previous question provides a reference to an article of Cohn (which I could only get a quarter of). However, he proves that there is no noetherian, artinian, or commutative (which we have already seen by different means) where (1) is negative.
The example of a ring which does not satisfy (1) that he gives does not have IBN.
But the cases in which he proves it affirmative are all classes of rings that have IBN! Can it be, then, that the answer to (1) is positive for every ring with IBN, or is there a ring with IBN where the answer to (1) is negative? (It would have to be a non noetherian, non artinian, noncommutative ring!)