In a Noetherian ring $R$ every idempotent ideal is generated by a single idempotent element of $R$, that is, $I=Re$. Suppose that, this time, the idempotent $e\in R$ is central and the idempotent ideal is $J=Re=eR$. Is there an example of a ring $R$ with many ideals $\{I_1,I_2,\ldots,I_n\}$ but only a few, say $\{I_1~\text{and}I_2\}$, have the property that $J=Re=eR$ and the rest, say $\{I_3,\ldots,I_n\}$, are not idempotent ideals? I am looking for examples even from non-Noetherian case.
2026-04-09 00:24:28.1775694268
Rings with fewer idempotent ideals generated by a single central idempotent.
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If $R$ is any nonfield integral domain, then every ideal except the trivial ones is not idempotent. Doesn’t that fit your description?