Definition 5

Let $B$ be a ring, $F$ a $B$-module to the left (resp. to the right) and $J$ an antiautomorphism of $B$. We denote by $F^J$ the $B$-module to the right (resp. to the left) having the same underlying additive group as $F$ and in which the external composition law is $(b, y) \to b^{J'}y$ (resp. $(b, y) \to yb^{J'}$) $(b \in B,\ y \in F,\ J' = J^{-1})$.

With the notations of Definition 5, a linear mapping from $F^J$ into a $B$-module to the right (resp. to the left) $H$ is therefore identified with a $\mathbf{Z}$-linear mapping $u$ from $F$ into $H$ satisfying

The mapping $u$ from $F$ into $H$ is a semilinear mapping from $F$ into $H$ relative to $J$ (chap. II, App. I, no 1), if one considers $J$ as an isomorphism of the ring $B^0$ opposite to $B$ onto $B$, and $F$ as a $B^0$-module to the right (resp. to the left).

Analogously a right sesquilinear mapping $\Phi$ (for $J$) from $E \times F$ into $G$, where $F$ is a $B$-module to the left, is identified with a bilinear mapping from $E \times F^J$ into $G$; if the latter is degenerate to the right (resp. degenerate to the left, non-degenerate), one says that $\Phi$ is degenerate to the right (resp. degenerate to the left, non-degenerate).