这是一份kcl伦敦大学学院 5CCM121B作业代写的成功案
Proof. Let $\varphi: C(B) \longrightarrow C(A)$ be a homomorphism of $K$-algebras. Let $q_{j}$ : $B \longrightarrow K,\left(y_{1}, \ldots, y_{n}\right) \longmapsto y_{j}$ be the $j$ th coordinate function of $B$. The polynomial function $B \longrightarrow \bar{K}$ induced by $g=\sum_{k} c_{k} Y_{1}^{k_{1}} \cdots Y_{n}^{k_{n}} \in K\left[Y_{1}, \ldots, Y_{n}\right]$ sends $y=\left(y_{1}, \ldots, y_{n}\right) \in B$ to
\begin{aligned}
g\left(y_{1}, \ldots, y_{n}\right) &=g\left(q_{1}(y), \ldots, q_{n}(y)\right) \
&=\sum_{k} c_{k} q_{1}(y)^{k_{1}} \ldots q_{n}(y)^{k_{n}}=\left(\sum_{k} c_{k} q_{1}^{k_{1}} \cdots q_{n}^{k_{n}}\right)(y)
\end{aligned}
5CCM121B COURSE NOTES ：
If $\varphi: A \longrightarrow B$ is a homomorphism of left R-modules, then $\varphi_{}=\operatorname{Hom}{R}(M, \varphi): \operatorname{Hom}{R}(M, A) \longrightarrow \operatorname{Hom}{R}(M, B)$ is a homomorphism of abelian groups. Moreover: (1) if $\varphi$ is the identity on $A$, then $\varphi{}$ is the identity on $\operatorname{Hom}{R}(M, A)$; (2) $(\psi \circ \varphi){}=\psi_{} \circ \varphi_{}$ whenever $\varphi: A \longrightarrow B$ and $\psi: B \longrightarrow C$; (3) $(\varphi+\psi){}=\varphi{}+\psi_{}$ whenever $\varphi, \psi: A \longrightarrow B$.
Similarly, when $M, N$, and $A$ are left $R$-modules, every module homomorphism $\varphi: M \longrightarrow N$ induces a mapping