Definition:Module Defined by Ring Homomorphism

Definition
Let $A$ and $B$ be a rings.

Let $f : A \to B$ be a ring homomorphism.

Definition 1
The left $A$-module structure of $B$ via $f$ is the module with left ring action:
 * $A \times B \to B$
 * $ \tuple {a, b} \mapsto \map f a \cdot b$

Definition 2
The left $A$-module structure of $B$ via $f$ is the restriction of scalars of the $B$-module structure of $B$.

Definition 3
Let $\lambda: B \to \map {\operatorname {End} } B$ be its left regular ring representation.

The left $A$-module structure of $B$ via $f$ is the module with ring representation the composition $\lambda \circ f$.

Also see

 * Equivalence of Definitions of Module Defined by Ring Homomorphism
 * Definition:Algebra Defined by Ring Homomorphism