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!