= Quotient of a finite product of fields
{title2=$(\prod_{j=1}^rK_j)/J\cong\prod_{j\in D}K_j$}
A <quotient ring> of a finite <direct product of rings> whose factors are <fields> is a product of a subset of those factors. By <ideals of a finite product of rings>, write $J=\prod_jJ_j$. Since a <field> has only the zero <ideal> and the whole <field> as <ideals>, every $J_j$ is one of these. The <ring homomorphism> $(x_j)\mapsto(x_j+J_j)$ is surjective with <kernel of a ring homomorphism> $J$, so the <first isomorphism theorem for rings> gives $(\prod_jK_j)/J\cong\prod_j(K_j/J_j)$. Discard the <zero ring> factors. If no factors survive, the result is the <zero ring>.
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