Remark

Let $A$ and $B$ be two rings, $J_1$ an antiautomorphism of $A$, $M$ an $A$-module to the right, $N$ a $B$-module to the right and G a (A, B)-bimodule. One says that a mapping $\Phi$ from $M \times N$ into G is left sesquilinear for $J_1$ if it is $\mathbf{Z}$-bilinear and if it satisfies

Such a mapping is identified with a bilinear mapping from $M^{J_1} \times N$ into G. We shall often leave to the reader the task of transposing to left sesquilinear mappings the definitions and properties given for right sesquilinear mappings; when we speak of a sesquilinear mapping (without further specification), it will be a right sesquilinear mapping.