---- > [!definition] Definition. ([[algebra]]) > Let $R$ be a [[commutative ring]]. An **$R$-algebra** is a [[ring homomorphism]] $\alpha:R \to S$ such that the [[subring]] $\alpha(R)$ is contained in the [[center of a ring|center]] of $S$.[^1] > > The usual abuse of language allows us to refer to an $R$-algebra by the name of the target $S$ (on which $\alpha$ [[module induced by a ring homomorphism|induces]] [[module]] structure) of the [[ring homomorphism|homomorphism]] $\alpha$. > > $R$-algebras are objects of [[category]] $R$-$\mathsf{Alg}$. The homset $\text{Hom}(S,T)$ is the set of [[algebra homomorphism|algebra homomorphisms]] between the [[ring|rings]] $S$ and $T$. > > Or we can view $R$-$\mathsf{Alg}$ as a [[subcategory]] of the [[coslice category]] $\mathsf{Ring}^{R}$, wherein we keep just the [[ring homomorphism|ring homomorphisms]] $\varphi$ *that respect scalar multiplication*. So the objects look like $R \xrightarrow{\alpha}S$ and morphisms are commutative diagrams as below: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAGAXVJADcBDAGwFcYkQAlEAX1PU1z5CKAEyli1Ok1bsAKjz4gM2PASLlxkhizaIQAZR6SYUAObwioAGYAnCAFskGkDghIxIRvQBGMRgAUBVWEQGyxTAAscEBptGT0AHQSmNAj6BWs7R0RnVyQyKR12JN8cdN5Mh3caPMQCuN0QJIYbVKwYzx8-QJUhdjDI6O5KbiA > \begin{tikzcd} > & R \arrow[ld, "\alpha"'] \arrow[rd, "\beta"] & \\ > S \arrow[rr, "\varphi"'] & & T > \end{tikzcd} > \end{document} > ``` > > ^definition (where? On any [[algebra|associative algebra]] $(A, \cdot)$ may be defined a [[Lie algebra|Lie bracket]], via the [[commutator|commutation]] $[x,y]:=x \cdot y - y \cdot x$. When viewing $A$ as a [[Lie algebra]] in this manner, we write $\text{Lie}(A)$. > [!basicexample] > If $A=\text{End }V$, $V$ a [[vector space]], then $\text{Lie}(A)=\mathfrak{gl}(V)$. ^basic-example ) > [!note] Remark. > Thus, $S$ is a set simultaneously endowed with addition (as it is an [[abelian group]]), multiplication (as it is a [[ring]]), and $R$-[[module]] structure ([[module induced by a ring homomorphism|induced]] by $\alpha$ via the map $(r,s)\mapsto \alpha(r)s$). Note that if $R=k$ is a [[field]], the structure morphism $k \to S$ is an embedding $(S \neq 0)$.[^1] ^note > [!basicexample] > - For $k$ a [[field]], then the [[ring]] of [[polynomial 4|polynomials]] is a $k$-[[algebra]]: $\begin{align} k &\to k[T_{1},\dots,T_{n}] \\ {\underbrace{a}_{\in k}} &\mapsto \underbrace{a}_{\text{constant polynomial}} \end{align}$ as is $\begin{align} k &\to k[T_{1},\dots,T_{n}] / I \\ a &\mapsto a + I. \end{align}$ >- Since $\mathbb{Z}$ is [[terminal object|initial]] in $\mathsf{Ring}$, every ring $S$ is a $\mathbb{Z}$-algebra in a unique way. ^basic-example [^1]: It is not quite right, though, to say 'a nonzero algebra over a field $k$ is just a [[ring]] that contains $k > [!justification] > The inspiration here comes from considering the [[module induced by a ring homomorphism]] $\alpha:R \to S$ in the case that $R$ is [[commutative ring|commutative]] and $\im \alpha \subset \text{center}(S)$ so that $\alpha(r),s$ commute for every $r \in R, s \in S$. Indeed, in this case the left/right module structures agree and the ring operation in $S$ $(s_{1}, s_{2}) \mapsto s_{1}s_{2}$ is compatible with the $R$-module structure in the sense that $(r_{1}s_{1})(r_{2}s_{2})=\alpha(r_{1})s_{1}\alpha(r_{2})s_{2}=\alpha(r_{1})\alpha(r_{2})s_{1}s_{2}=(r_{1}r_{2})(s_{1}s_{2}).$ So to get 'bilinear multiplication' we need $\alpha(R)$ to lie in the center of $S$. ^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 > ``` . This definition is incompatible with the notion of a $k$-[[algebra homomorphism]]. > [!justification] > The inspiration here comes from considering the [[module induced by a ring homomorphism]] $\alpha:R \to S$ in the case that $R$ is [[commutative ring|commutative]] and $\im \alpha \subset \text{center}(S)$ so that $\alpha(r),s$ commute for every $r \in R, s \in S$. Indeed, in this case the left/right module structures agree and the ring operation in $S$ $(s_{1}, s_{2}) \mapsto s_{1}s_{2}$ is compatible with the $R$-module structure in the sense that $(r_{1}s_{1})(r_{2}s_{2})=\alpha(r_{1})s_{1}\alpha(r_{2})s_{2}=\alpha(r_{1})\alpha(r_{2})s_{1}s_{2}=(r_{1}r_{2})(s_{1}s_{2}).$ So to get 'bilinear multiplication' we need $\alpha(R)$ to lie in the center of $S$. ^justification ---- #### ---- #### References > [!backlink] > {CODE_BLOCK_PLACEHOLDER} > [!frontlink] > {CODE_BLOCK_PLACEHOLDER}