A criterion for the properness of star-transposition as an involution on n × n matrices over any associative star-ring

On the ring Mn(C) of all n × n complex matrices A, the conjugate transpose involution A↦A* has the "properness" property A*A=0⇒A=0, as required for the existence of the partial order known as the "*-order" on Mn(C). More generally, related questions are asked and answered about the ring Mn(R) of all n × n matrices over an arbitrary associative ring R with any given involution R→R (as a generalization of the conjugacy map C→C), yielding a corresponding "*-transposition" involution on Mn(R). A criterion is found for this involution of Mn(R) to be proper, so that Mn(R) has a corresponding *-order. The special case R=Zk is also considered.