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

Abstract

On the ring $M_n(\mathbb{C})$ of all n × n complex matrices A, the conjugate transpose involution $A\mapsto A^{*}$ has the “properness” property $A^{*}A=0\Rightarrow A=0$, as required for the existence of the partial order known as the “$*$-order” on $M_n(\mathbb{C})$. More generally, related questions are asked and answered about the ring $M_n(R)$ of all n × n matrices over an arbitrary associative ring R with any given involution $R\to R$ (as a generalization of the conjugacy map $\mathbb{C}\to \mathbb{C}$), yielding a corresponding “$*$-transposition” involution on $M_n(R)$. A criterion is found for this involution of $M_n(R)$ to be proper, so that $M_n(R)$ has a corresponding $*$-order. The special case $R={\mathbb{Z}}_k$ is also considered.

KEYWORDS:

• Associative ring
• involution
• matrix ring
• partial order
• properness
• star-order

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 06A06
• 16W10