Idempotents and product decompositions

Idempotents and product decompositions: Let $R$ be a ring and let $e\in R$ be a central idempotent (so $e^2=e$ and $er=re$ for all $r\in R$). Then $eR$ and $(1-e)R$ are two-sided ideals, and the map

is a ring isomorphism with inverse $(a,b)\mapsto a+b$. Conversely, any product decomposition $R\cong A\times B$ determines a central idempotent corresponding to $(1,0)$.

A central idempotent in the center splits a ring into a product of rings via complementary ideals . This mechanism is closely related to Chinese remainder decompositions and is often used to build explicit splittings from orthogonal idempotents.