Timeline for Another question about primitive central idempotents in associative unital rings (yes, again!)
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Feb 6, 2011 at 20:04 | comment | added | arsmath | 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. | |
| Feb 6, 2011 at 19:40 | vote | accept | ingrem | ||
| Feb 6, 2011 at 19:40 | comment | added | ingrem | @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. | |
| Feb 6, 2011 at 19:37 | comment | added | darij grinberg | If you want to rescue the question, try restricting to some "nice" rings (Noetherian or Artinian). | |
| Feb 6, 2011 at 19:37 | comment | added | darij grinberg | In my case, yes, but not generally. Unless your ring has a topology. | |
| Feb 6, 2011 at 19:26 | comment | added | ingrem | @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? | |
| Feb 6, 2011 at 19:05 | history | answered | Steven Landsburg | CC BY-SA 2.5 |