Let $R$ be an arbitrary associative unital ring and let $x \in R$. Can we always represent $x$ as a finite or infinite sum $x=\sum\_{i}y\_ie\_i$ where $y\_i\in R$ and each $e\_i$ is a primitive central idempotent in $R$?


P.S. Sorry if the question is stupid. Last 10 years ring theory was not my field of specialization, but today I should understand my own (really old) article about generic norms of associative finite-dimensional algebras. And some things, which were obvious for me 10 years ago, are not obvious anymore. Thanks for your answers!



Another question about primitive central idempotents in associative unital rings (yes, again!)
I have no idea what an infinite sum means in an arbitrary associative unital ring.


Your question is equivalent to asking whether it is possible to write 1 as a sum of primitive central idempotents. (If $1=e\_1+\ldots+e\_n$, then you can take $y\_i=x$ for all $i$.)


