---- > [!definition] Definition. ([[algebra homomorphism]]) > Let $R$ be a [[commutative ring]] and $\alpha:R \to S$ an $R$-[[algebra]]. [[module induced by a ring homomorphism|This places]] $R$-[[module]] structure on the [[ring]] $S$. If $\beta:R \to T$ is another $R$-[[algebra]], then an **$R$-algebra homomorphism** $\varphi:S \to T$ is a map which is both a [[ring homomorphism]] and a [[linear map]]: $\begin{align} \varphi(s_{1}s_{2})=&\varphi(s_{1})\varphi(s_{2}) \\ \varphi(s_{1}+s_{2})=& \varphi(s_{1}) + \varphi(s_{2}) \\ \varphi(1_{S})=&1_{T}\\ \varphi(rs)=&r \varphi(s) \end{align}$ holds for all $s_{1},s_{2} \in S$ and $r \in R$. > > > Put differently, $\varphi$ respects the two $R$-algebra structures in the sense that the following diagram commutes: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAGAXVJADcBDAGwFcYkQAlEAX1PU1z5CKcqWLU6TVuwDKPPiAzY8BIgCYxEhizaIQAFR4SYUAObwioAGYAnCAFskokDghIyknewA63mwAsIAH05GkZ6ACMYRgAFARVhEBssU38ceWs7R0RnVyQNT2k9XwDgw15Mh3caPMQC7SKQXwYbNH8sEDDI6LjlIXZk1PTuSm4gA > \begin{tikzcd} > & R \arrow[ld, "\rho_S"'] \arrow[rd, "\rho_T"] & \\ > S \arrow[rr, "\varphi"'] & & T > \end{tikzcd} > \end{document} > ``` > > recalling $\rho_{S}(r)=r \cdot 1_{S}$, this is concisely captured by the assertion $\varphi(r \cdot 1_{S}) = \varphi \cdot 1_{T} \text{ for each } r \in R.$ > [!equivalence] > [[upgrading a linear map to an algebra homomorphism]] ^equivalence ---- #### ---- #### 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 > ```