Definition 4

Let E and F be two (left) modules with respect to the same ring A. A linear mapping (or A-linear mapping, or homomorphism, or A-homomorphism) of E into F is any mapping u : E → F such that:

(3) $u(x + y) = u(x) + u(y)$ for $x \in E, y \in E;$

(4G) $u(\lambda . x) = \lambda . u(x)$ for $\lambda \in A, x \in E.$

If E and F are two right A-modules, a linear mapping $u : E \to F$ is a mapping satisfying (3) and

(4D) $u(x . \lambda) = u(x) . \lambda$ for $\lambda \in A, x \in E.$