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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!

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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 $i$.)

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  • $\begingroup$ @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? $\endgroup$
    – ingrem
    Commented Feb 6, 2011 at 19:26
  • $\begingroup$ In my case, yes, but not generally. Unless your ring has a topology. $\endgroup$
    – darij grinberg
    Commented Feb 6, 2011 at 19:37
  • $\begingroup$ If you want to rescue the question, try restricting to some "nice" rings (Noetherian or Artinian). $\endgroup$
    – darij grinberg
    Commented Feb 6, 2011 at 19:37
  • $\begingroup$ @darij grinberg: So, there is no sense to consider infinite sums in a general case (an arbitrary associative unital ring). Thanks in advance to all :) I'll use that strange "V" button to make the Steven's answer green. $\endgroup$
    – ingrem
    Commented Feb 6, 2011 at 19:40
  • $\begingroup$ The infinite sum is not an intrinsic part of the structure of the ring. For example, a ring homomorphism is not required to preserve the infinite sum. $\endgroup$
    – arsmath
    Commented Feb 6, 2011 at 20:04

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