Question

Let $R$ be an associative ring with identity and let $x$ be an arbitrary element from the ring $R$. Could you please help me to prove that $x=ye$, where $y$ is some element in $R$ and $e$ is some primitive central idempotent in $R$. In other words, I need to prove that any element in $R$ is representable as a product of some element and some primitive central idempotent in $R$.

Thanks for the answers! Answers like "you are not right, for example ..." and "see /book/, p. /page number/" are also OK.

Answer 1

Consider the direct product $k\times k$ of two copies of some field and the element $x=(1,1)$. In this simple example you can describe all primitive central idempotents.