Rings over which the transpose of every invertible matrix is invertible. (Q734791)

The authors prove that the transpose of every invertible square matrix over a ring \(R\) is invertible if and only if \(R/\text{rad}(R)\) is commutative. Many other characterizations are obtained for such rings. They also show that, for von Neumann regular rings, this is a necessary and sufficient condition for the commutativity of \(R\).