Remark

If the groups of the diagram (10) are all (for example, right) modules over a ring $A$ and the homomorphisms are A-module homomorphisms, it is soon verified that the homomorphism d defined in Proposition 2, (iii) is also a A-module homomorphism: if $x \in \mathrm{Ker}(c)$ and $a \in A$, and $y \in B$ is such that $v(y) = k(x)$, it is sufficient to note that $v(ya) = k(xa)$.