Definition. central involution in an algebra [fgap-000W]
Definition. central involution in an algebra [fgap-000W]
Let \(R\) be a commutative ring, let \(A\) be an associative unital \(R\)-algebra, and suppose \(2\cdot 1_R\) is invertible. Write \[ h=(2\cdot 1_R)^{-1}. \] A central involution for this construction is an element \(z\in A\) such that \[ z^2=1_A,\qquad za=az\quad \text {for every }a\in A. \] We then define \[ e_+=h(1_A+z),\qquad e_-=h(1_A-z), \] using the structural map \(R\to A\) for the scalar \(h\).
Only the equation \(z^2=1_A\) is used. The element need not have exact order 2: the extra condition \(z\neq 1_A\) merely excludes the degenerate case \(e_-=0\). The algebra \(A\) need not be commutative.