Remark

Suppose that the algebra A is without radical. Let $\boldsymbol{a}= (a_1, . . . , a_n)\in A^n$. By Proposition 8 of I, p. 38, the mapping $\Theta^U_{\boldsymbol{a}}$ is the unique mapping $\varphi$ from $\mathscr{O}(U)$ into A such that $\chi (\varphi (f)) =f(\chi (a_1), . . . , \chi (a_n))$ for every $\chi \in \mathsf{X}(A)$ and every function $f\in \mathscr{O}(U)$.