Remark

When E and F are two commutative groups considered as modules over the ring $\mathbf{Z}$ (no. 1), every homomorphism $u$ of the group E (without operators) into the group F (without operators) is also a linear mapping of E into F, since for n an integer > 0, the relation $u(n . x) = n . u(x)$ follows from $u(x + y) = u(x) + u(y)$ by induction on n and for $n = -m < 0$,