[Math] Given a ring with unity and a central idempotent element e, prove some isomorphic relations

Question

Given a ring $R$ with $1\neq 0$, and an element $e$ that is idempotent which is central in $R$, I want to prove that $R/Re \cong R(1-e)$, $R/R(1-e)\cong Re$, and subsequently, $R\cong Re\times R(1-e)$. My intuition is pointing me to the isomorphism theorems, and the last should follow from the Chinese remainder theorem, but I'm at a loss as to how to connect the dots.

Answer 1

You can prove directly that $R\cong Re\times R(1-e)$. The natural map works:

Let $f=1-e$. Then $e+f=1$ and $ef=0=fe$. Consider $\phi: R\to Re\times R(1-e)$ given by $\phi(x)=(xe,xf)$. it is easy to see that $\phi$ is injective. To prove that it is surjective, prove that $\phi(ye+zf)=(ye,zf)$.