Let $R$ be a (not necessarily) commutative ring and $e \in R$ some idempotent. Then the Pierce decomposition writes $$ eRe \oplus (1-e)Re \oplus eR(1-e) \oplus (1-e)R(1-e). $$ I tried to construct such an isomorphism, specifically consider the cartesian product $eRe \times eR(1-e) \times (1-e)Re \times (1-e)R(1-e)$, and the mapping $$ \varphi(r) = (ere, re - ere, er - ere, r - er - (re - ere)). $$ Then this is bijective. And also it is an isomorphism of $(R, +)$ and the additive group we get by componentwise addition on this cartesian product.
But the only case that multiplication makes any sense is when $e$ is central, i.e. commutes with all elements, otherwise I am not even able to define multiplication componentwise in the above case. But the Pierce decomposition, for example also here and in the above link, does not require the idempotents to be central. So I am unable how this decomposition should work out, and I am surprised by the above inconsistencies? So could anyone explain them?
Also for the preservation of the additive structure, and surjectivity and injectivity of $\varphi$ nowhere is it needed that we have $e^2 = e$.