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Let $R$ be a (possibly noncommutative) ring. Recall that an ideal $I \subseteq R$ is called idempotent if $I^2=I$, meaning that every element of $I$ can be written as a finite sum of products of elements from $I$.

Suppose that every two-sided maximal ideal $M \subseteq R$ is idempotent: that is, $M^2=M$ for all two-sided maximal ideals $M$.

Question:

  • Under this assumption, what can we say about the structure of the ring $R$ ?
  • In particular, does this condition imply that $R$ must be semisimple Artinian (for example, a finite product of simple rings)?
  • Are there interesting examples or counterexamples if $R$ is not assumed to be Artinian?

Any references, examples, or structural results would be appreciated.

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    $\begingroup$ Cross-posted on math.stackexchange $\endgroup$ Commented Apr 27 at 8:46
  • $\begingroup$ For commutative $R$, this holds iff every module of finite length is semisimple. (The proof of Th. 2.15 shows that it is in fact sufficient that every module of length 2 is semisimple.) $\endgroup$ Commented Apr 29 at 14:27

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In a von Neumann regular ring, every two sided ideal is idempotent. This is a large class of rings. It contains many that are not semisimple Artinian.

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  • $\begingroup$ However the condition of the OP is not enough to get von Neumann regular. See math.stackexchange.com/questions/1678681/… $\endgroup$ Commented Apr 26 at 18:52
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    $\begingroup$ No, I'm aware of that. For example, there are rings with no maximal ideals at all, such as indiscrete valuation rings. These vacuously satisfy the hypothesis. $\endgroup$ Commented Apr 26 at 18:54
  • $\begingroup$ Yes I know but I would like to find characterizations of rings where every two sided maximal ideal is idempotent. $\endgroup$ Commented Apr 26 at 18:55
  • $\begingroup$ I guess I was only answering one of your questions. $\endgroup$ Commented Apr 26 at 19:00
  • $\begingroup$ @DaveBenson I don't think what you wrote in your first comment is what you meant to write. But in a valuation ring whose value group has no minimum positive element, the unique maximal ideal is idempotent. $\endgroup$ Commented Apr 27 at 18:12

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