Involutive Ring

An involutive ring is a Ring $R$ equipped with an involutive automorphism $(-)^{\star}:R^{\mathrm{op}}\to R$ .

Explicitly, this means that we have:

Basic Properties

Note that we can derive $1^{\star}=1$ via the following chain of reasoning:

$\displaystyle 1^{\star}$	$\displaystyle=1^{\star}\cdot 1$	(right id)
	$\displaystyle=1^{\star}\cdot(1^{\star})^{\star}$	(involution)
	$\displaystyle=(1\cdot 1^{\star})^{\star}$	(homomorphism)
	$\displaystyle=(1^{\star})^{\star}$	(left id)
	$\displaystyle=1$	$\displaystyle\square$

A similar argument lets us derive that $0^{\star}=0$ .

We can also deduce that $(x^{-1})^{\star}=(x^{\star})^{-1}$ by the usual argument that group homomorphisms preserve inverses.