TITLE: The A-valued points on a scheme
QUESTION [6 upvotes]: Let $A$ be a ring and $X=Spec(\mathbb{Z}[x_1,...,x_n] / (f_1,...,f_r))$ an affine scheme where $f_i \in \mathbb{Z}[x_1,...,x_n]$. The $A$-valued points of $X$ are defined to be morphisms $Spec(A)\rightarrow X$. Show that the $A$ valued points of $X$ are the solutions to $f_1(x_1,...,x_n)=\cdots=f_r(x_1,...,x_n)=0$ in the ring $A$.

REPLY [2 votes]: A morphism of affine schemes $\operatorname{Spec} A\to \operatorname{Spec} R$ is induced by a ring morphism $R\to \Gamma(\mathscr{O}_A,\operatorname{Spec} A) = A$ by the adjoint equivalence of categories between $\mathfrak{Ring}$ and $\mathfrak{Aff}^{opp}$ given by $(\Gamma,\operatorname{Spec})$.
A morphism $\pi':\mathbb{Z}[x_1,\cdots, x_n]/(f_1,\cdots, f_r)\to A$ is the same as a morphism of $\pi:\mathbb{Z}[x_1,\cdots, x_n]\to A$ such that $(f_1,\cdots, f_r)$ vanishes, which is the same as points $a_i = \pi(x_i)$ in $A$ such that all $f_j(a_1,\cdots, a_n)$  vanishes.