# Another question about primitive central idempotents in associative unital rings (yes, again!)

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!

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 $i$.)
• @Steven Landsburg: Thanks for the answer. Yes, you are right, if the sum is finite, then we are talking about Pierce decomposition by a complete set of orthogonal central primitive idempotents. About an infinite sum: suppose $R$ is the ring of all countable sequences of rationals, with coordinatewise addition and multiplication (thanks darij grinberg for the example). The identity of $R$ is the infinite sum $(1,1,1,\ldots) = (1,0,0,\ldots) + (0,1,0,\ldots) + (0,0,1,\ldots)+\ldots$, is it? If so, we probably can talk in this sense about infinite sums in associative unital rings. No? – ingrem Feb 6 '11 at 19:26