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Let $R$ be a (commutative or non-commutative) ring with identity. As usual, an idempotent $e \in R$ is primitive if $eR$ (the principal right ideal generated by $e$) is indecomposable as a right module over $R$; equivalently, $e \ne 0_R$ and $e \ne f+g$ for all non-zero orthogonal idempotents $f, g \in R$.

Question. Is there a standard name and/or a "non-trivial characterization" for the rings $R$ with the property that, whenever $e$ and $f$ are primitive idempotents, then the right $R$-modules $eR$ and $fR$ are isomorphic?

The condition is equivalent to requiring that any two indecomposable direct summands of the regular right module $R_R$ are isomorphic.

For instance, every von Neumann regular ring satisfying the comparability axiom (as per the unnumbered definition on p. 80 of Goodearl's book on von Neumann regular rings) fulfills the above condition: In particular, this is the case with any prime, von Neumann regular, right self-injective ring (loc. cit., Corollary 9.16).

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  • $\begingroup$ For an Artinian ring this is the same as being a matrix algebra over a local ring. I guess this should also be true for a semiperfect ring $\endgroup$ Jun 30, 2021 at 19:49

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