Idempotents and product decompositions

Idempotents and product decompositions: Let $R$ be a unital 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

R\longrightarrow eR\times (1-e)R,\qquad r\longmapsto (er,(1-e)r)

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 $(1,0)$ .

Remarks

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.