# Show that R can be written as a direct product of two or more (nonzero) rings iff R cont

Alaina Marshall 2022-05-24 Answered
Show that $R$ can be written as a direct product of two or more (nonzero) rings iff $R$ contains a non-trivial idempotent. Show that if $e$ is an idempotent, then $R=Re×R\left(1-e\right)$ and that $Re$ may be realized as a localization, $Re=R\left[{e}^{-1}\right]$.
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nicoupsqb
Assume $A\cong {A}_{1}\oplus {A}_{2}$. It follows $e:=\left(1,0\right)\ne 1$ is a non trivial idempotent. Assume conversely that $1\ne e\in A$ is a non trivial idempotent. Let $I:=\left(e\right),J:=\left(e-1\right)\subseteq A$ be the ideal generated by $e,e-1$. It follows $I+J=\left(1\right)$ hence these ideals are coprime. Moreover $IJ=I\cap J$. Since $IJ:=A\left(e\left(e-1\right)\right)=A\left({e}^{2}-e\right)=A\left(0\right)=\left(0\right)$ is the zero ideal, we get by the CRT an isomorphism is the zero ideal, we get by the CRT an isomorphism
$A\cong A/IJ\cong A/I×A/J$
where $A/I,A/J\ne 0$. Hence by the CRT we get that $A$ has a non-trivial idempotent iff $A$ may be written as a (non-trivial) direct sum of two rings.