Invertible linear maps preserving \(\{1\}\)-inverses of matrices over PID
Let \(R\) be a principal ideal domain of characteristic two. The author classifies linear bijections \(f:M_n(R)\to M_n(R)\) with the property that \(AXA=A\) implies \(f(A)f(X)f(A)=f(A)\). The proof first shows that \(f\) is unital, and then uses the fact that, in characteristic two, \(B^2=B\) is equivalent to \(B^3=B\) and \((B+I)^3=(B+I)\). Consequently, \(f\) preserves idempotents.
scientific article; zbMATH DE number 5119547