# 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 $\lambda \circ f$.