---- > [!definition] Definition. ([[module induced by a ring homomorphism]]) > Any [[ring homomorphism]] $\alpha:R \to S$ may be used to define an interesting (left-) $R$-[[module]] via the action $\begin{align} R \times S \to& S \\ (r,s) \mapsto& \alpha(r)s, \end{align}$ where the [[binary operation|operation]] on the right is simply multiplication in $S$. > >It is common to write $rs$ rather than $\alpha(r)s$. ^definition > [!justification] > It is immediate from the [[ring|ring axioms]] and the fact that $\alpha$ is a [[ring homomorphism]] that this indeed places a module structure on $S$. ^justification ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```