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