Remark 8

Let $E, E_1$ be two $A$-modules, $\Phi$ (resp. $\Phi_1$) a sesquilinear form on $E$ (resp. $E_1$). Suppose that $\Phi_1$ is non-degenerate and that there exist an element $\alpha \in A$ and a bijection $u$ of $E$ onto $E_1$ such that $\Phi_1(u(x), u(y)) = \Phi(x, y)\alpha$ for all $x, y$ in $E$. Show that: 1° $\Phi$ is non-degenerate ; 2° $u$ is linear ; 3° if $E_1$ is a faithful $A$-module, the same is true of $E$, and $\alpha$ is not a right divisor of 0 in $A$ ; 4° if $\Phi_1$ takes values in $A$ which are not left divisors of 0, the same is true of $\Phi$.