Let $R$ be an associative ring with unity $1$ generated by all its idempotents. Denoted by $R^n$ the additive subgroup of $R$ generated by all elements $a_1a_2\cdots a_n$ for $a_i\in R$. Then, is $R=R^n$ true? I initially consider $n=2$. I think this is true but I still feel confused. Please help me consider the situation. Any counterexample or reference or technique is very much appreciated. Thank you in advance. On the other hand, please let me know any results related to such a ring if you know.
2026-03-28 06:59:06.1774681146
Ring generated by all its idempotents
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