Definition:Module Defined by Ring Homomorphism

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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