Remark

Let A be a ring and $\sigma$ an anti-endomorphism of A (no. 3); consider the two following situations:

(GD) E is a left A-module, F a right A-module and $u$ a $\mathbf{Z}$-linear mapping of E into F such that $u(ax) = u(x)\sigma(a)$ for $a \in A,\ x \in E$; in other words, $u$ is a semi-linear mapping relative to $\sigma$ of the right $A^0$-module E into the right A-module F.

(DG) E is a right A-module, F a left A-module and $u$ a $\mathbf{Z}$-linear mapping of E into F such that $u(xa) = \sigma(a)u(x)$ for $a \in A,\ x \in E$; in other words, $u$ is a semi-linear mapping relative to $\sigma$ of the left $A^0$-module E into the left A-module F.

In the two cases, the matrix $M(u)$ of $u$ relative to bases of E and F has its elements in A; if these bases are finite, then, for all $x \in E$, we have the respective formulae

(17 GD)

(17 DG) the products on the two sides being calculated in A. This follows immediately from (26 D) and (26 G) respectively.