The spectrum of a finite product ring is the disjoint union of the factor spectra

Each $e_i$ is an idempotent, $e_i{e}_j=0$ for $i\mathrm{\ne }j$ and $e_1+\cdots +e_r=1_R$, since these are componentwise computations in the product ring; in particular $1-e_i={\sum }_{j\mathrm{\ne }i}{e}_j$ is again an idempotent.

The projection ${\pi }_i$ is a surjective unital ring homomorphism with kernel $I_i=\{a\in R:a_i=0\}$, and $I_i=(1-e_i)R$: every $(1-e_i)a$ has $i$-th coordinate $0$, and conversely every $a$ with $a_i=0$ equals $(1-e_i)a$. Also ${\pi }_i(e_i)=1_{R_i}$.

Let $e\in A$ be an idempotent and let $\mathfrak{q}\subseteq A$ be a prime ideal. Then $e\mathrm{\notin }\mathfrak{q}$ if and only if $1-e\in \mathfrak{q}$: since $e(1-e)=0\in \mathfrak{q}$, primality gives $e\in \mathfrak{q}$ or $1-e\in \mathfrak{q}$, and both cannot occur because then $1=e+(1-e)\in \mathfrak{q}$, contradicting the properness of a prime ideal.

Let $\mathfrak{q}\subseteq R$ be an ideal with $I_i\subseteq \mathfrak{q}$ and let $\mathfrak{p}={\pi }_i(\mathfrak{q})\subseteq R_i$ be its image. Then $\mathfrak{q}={\pi }_i^{-1}(\mathfrak{p})$ and $R\mathrm{/}\mathfrak{q}\cong R_i\mathrm{/}\mathfrak{p}$: the quotient map $R\to R\mathrm{/}{I}_i$ identifies $R\mathrm{/}{I}_i$ with $R_i$ and carries $\mathfrak{q}$ to $\mathfrak{p}$, so the correspondence of ideals and the first isomorphism theorem give both statements.

Every prime ideal $\mathfrak{q}\subseteq R$ contains $e_j$ for all but exactly one index $j$: if two distinct elements $e_i,e_j$ both lay outside $\mathfrak{q}$, then $e_i{e}_j=0\in \mathfrak{q}$ would force one of them into $\mathfrak{q}$ by primality; and if all $e_i$ lay in $\mathfrak{q}$, then $1=e_1+\cdots +e_r\in \mathfrak{q}$, contradicting properness.

The principal localisation ${\lambda }_i:R\to R_{e_i}$ and the quotient map $q_i:R\to R\mathrm{/}{I}_i$ are both localisations of $R$ at the multiplicative set $S_i=\{1,e_i\}=\{e_i^n:n\ge 0\}$: each sends $e_i$ to a unit, and every unital ring map $\phi :R\to T$ with $\phi (e_i)$ a unit satisfies $\phi (1-e_i)=0$, because $\phi (e_i)\phi (1-e_i)=0$ and $\phi (e_i)$ is invertible, so $(1-e_i)R=I_i\subseteq \ker \phi$ and $\phi$ factors uniquely through $q_i$ by the quotient universal property. By uniqueness of localisations there is therefore a unique ring isomorphism ${\mathrm{\Theta }}_i:R_{e_i}\to R\mathrm{/}{I}_i$ with ${\mathrm{\Theta }}_i{\lambda }_i=q_i$, and composing with the canonical isomorphism $R\mathrm{/}{I}_i\cong R_i$ of step 1.4 gives a ring isomorphism, again written ${\mathrm{\Theta }}_i$, satisfying ${\mathrm{\Theta }}_i{\lambda }_i={\pi }_i$.

Let $\mathfrak{q}\subseteq R$ be a prime ideal with $e_i\mathrm{\notin }\mathfrak{q}$. Then $1-e_i\in \mathfrak{q}$ by step 1.3, so $I_i=(1-e_i)R\subseteq \mathfrak{q}$, and step 1.4 applied to $\mathfrak{p}={\pi }_i(\mathfrak{q})$ gives $\mathfrak{q}={\pi }_i^{-1}(\mathfrak{p})$ and $R\mathrm{/}\mathfrak{q}\cong R_i\mathrm{/}\mathfrak{p}$. Since $R\mathrm{/}\mathfrak{q}$ is an integral domain, so is $R_i\mathrm{/}\mathfrak{p}$, and therefore $\mathfrak{p}$ is a prime ideal of $R_i$.

Conversely, if $\mathfrak{p}\subseteq R_i$ is a prime ideal, then ${\pi }_i^{-1}(\mathfrak{p})$ is a prime ideal of $R$ with ${\pi }_i({\pi }_i^{-1}(\mathfrak{p}))=\mathfrak{p}$ and $e_i\mathrm{\notin }{\pi }_i^{-1}(\mathfrak{p})$: the composite $R\to R_i\to R_i\mathrm{/}\mathfrak{p}$ is a surjective ring homomorphism with kernel ${\pi }_i^{-1}(\mathfrak{p})$, so $R\mathrm{/}{\pi }_i^{-1}(\mathfrak{p})\cong R_i\mathrm{/}\mathfrak{p}$ is an integral domain and ${\pi }_i^{-1}(\mathfrak{p})$ is prime, the image statement holds because ${\pi }_i$ is surjective, and ${\pi }_i(e_i)=1_{R_i}\mathrm{\notin }\mathfrak{p}$.

For every index $i$ one has $D(e_i)=V((1-e_i))$, because by step 1.3 a prime $\mathfrak{q}$ satisfies $e_i\mathrm{\notin }\mathfrak{q}$ exactly when $1-e_i\in \mathfrak{q}$, and $V((1-e_i))$ is the set of primes containing the principal ideal $(1-e_i)$. Consequently each $D(e_i)$ is open, being a distinguished open, and closed, being a vanishing set, hence clopen.

The prime ideals of $R$ are exactly the ideals ${\pi }_i^{-1}(\mathfrak{p})$ with $i\in \{1,\dots ,r\}$ and $\mathfrak{p}\subseteq R_i$ prime, and each of them determines the pair $(i,\mathfrak{p})$ uniquely: existence and primeness are step 2.4, while a prime $\mathfrak{q}$ equals ${\pi }_i^{-1}(\mathfrak{p})$ for the unique index $i$ with $e_i\mathrm{\notin }\mathfrak{q}$ supplied by step 2.1 and $\mathfrak{p}={\pi }_i(\mathfrak{q})$, by step 2.3; and pairs with different indices give different primes, since ${\pi }_i^{-1}(\mathfrak{p})$ contains $e_j$ for $j\mathrm{\ne }i$ but not $e_i$, whereas ${\pi }_j^{-1}({\mathfrak{p}}^{\mathrm{\prime }})$ contains $e_i$ but not $e_j$.

The sets $D(e_1),\dots ,D(e_r)$ are pairwise disjoint and cover $\mathrm{Spec}R$: a prime $\mathfrak{q}$ lies in $D(e_i)$ exactly when $e_i\mathrm{\notin }\mathfrak{q}$, and by step 2.1 this holds for exactly one index.

The morphism $\mathrm{Spec}({\pi }_i):\mathrm{Spec}{R}_i\to \mathrm{Spec}R$ induced by ${\pi }_i$ is an isomorphism of locally ringed spaces onto the open locally ringed subspace $D(e_i)$ of $\mathrm{Spec}R$: by step 2.2 one has ${\pi }_i={\mathrm{\Theta }}_i\circ {\lambda }_i$ with ${\mathrm{\Theta }}_i$ a ring isomorphism, so functoriality gives $\mathrm{Spec}({\pi }_i)=\mathrm{Spec}({\lambda }_i)\circ \mathrm{Spec}({\mathrm{\Theta }}_i)$, where $\mathrm{Spec}({\mathrm{\Theta }}_i)$ is an isomorphism of locally ringed spaces and the morphism induced by ${\lambda }_i$ is identified with the open locally ringed subspace $D(e_i)$ by the principal-localisation description.

For $\mathfrak{q}={\pi }_i^{-1}(\mathfrak{p})$ the isomorphism of step 3.3 induces an isomorphism of local rings ${\mathcal{O}}_{\mathrm{Spec}R,\mathfrak{q}}\cong {\mathcal{O}}_{\mathrm{Spec}{R}_i,\mathfrak{p}}$, which the stalk formula further identifies with $(R_i{)}_{\mathfrak{p}}$; in particular the local rings of $\mathrm{Spec}R$ are exactly those of the factor spectra.

For $\mathfrak{q}={\pi }_i^{-1}(\mathfrak{p})$ step 1.4 gives $R\mathrm{/}\mathfrak{q}\cong R_i\mathrm{/}\mathfrak{p}$, so $\mathfrak{q}$ is maximal in $R$ exactly when $R_i\mathrm{/}\mathfrak{p}$ is a field, that is, exactly when $\mathfrak{p}$ is maximal in $R_i$; together with step 3.1 this describes all the maximal ideals of $R$.

Restricting sections to the pairwise disjoint clopen pieces gives a ring homomorphism $\rho :\mathrm{\Gamma }(\mathrm{Spec}R,\mathcal{O})\to {\prod }_{i=1}^r\mathrm{\Gamma }(D(e_i),\mathcal{O})$, and the sheaf axioms show that $\rho$ is bijective: it is injective because the $D(e_i)$ cover $\mathrm{Spec}R$, so a section is determined by its restrictions, and it is surjective because sections over the pieces are compatible on the empty overlaps, a sheaf having exactly one section over the empty set, and therefore glue to a global section. Composing $\rho$ with the isomorphisms $\mathrm{\Gamma }(D(e_i),\mathcal{O})\cong \mathrm{\Gamma }(\mathrm{Spec}{R}_i,\mathcal{O})$ induced by step 3.3 and with the canonical isomorphisms $\mathrm{\Gamma }(\mathrm{Spec}{R}_i,\mathcal{O})\cong R_i$ gives an isomorphism $\mathrm{\Gamma }(\mathrm{Spec}R,\mathcal{O})\cong {\prod }_{i=1}^r{R}_i$.

The $i$-th component of the isomorphism of step 4.3 is the composite of the restriction $\mathrm{\Gamma }(\mathrm{Spec}R,\mathcal{O})\to \mathrm{\Gamma }(D(e_i),\mathcal{O})$, which is the canonical localisation $R\to R_{e_i}$, with the isomorphism ${\mathrm{\Theta }}_i$ of step 2.2; since ${\mathrm{\Theta }}_i{\lambda }_i={\pi }_i$, this component is ${\pi }_i$ under the canonical identifications $\mathrm{\Gamma }(\mathrm{Spec}R,\mathcal{O})\cong R$ and $\mathrm{\Gamma }(\mathrm{Spec}{R}_i,\mathcal{O})\cong R_i$, so the isomorphism of step 4.3 is the identity of $R={\prod }_{i=1}^r{R}_i$. In particular the global sections of the disjoint union ${\coprod }_{i=1}^r\mathrm{Spec}{R}_i$ are ${\prod }_{i=1}^r{R}_i$, as asserted.

Claim 1 is step 1.1, claim 2 is step 3.1, claim 3 is steps 2.5, 3.2, 3.3 and 4.1, claim 4 is step 4.2, and claim 5 is steps 4.3 and 5.1; no step selects an element from a family, so the argument uses no choice.