Rings over which the transpose of every invertible matrix is invertible

We prove that the transpose of every invertible square matrix over a ring R is invertible if and only if R/rad(R) is commutative. Many other characterizations are obtained for such rings R in terms of U(R) (the group of units of R), including, for instance, c+ba∈U(R)⇒c+ab∈U(R), and 1+abc−cba∈U(R) (for all a,b,c∈R). We also consider a natural weakening of these conditions, namely, 1+abc∈U(R)⇒1+cba∈U(R), and show that, for von Neumann regular rings, this is a (necessary and) sufficient condition for the commutativity of R.