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Let $R$ be a ring and $I\subset R$ an ideal. Let's give an example of a ring $R$ such that $I^2=I$ but $I$ is not a direct summand of $R$.

Given ring $R$ and ideal $I$, i know that if $I$ is a direct summand of $R$ then $I^2=I$ but i can't show the converse.

user26857
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Desunkid
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  • Is your ring assumed to be unital? The claim that, if $I$ is a direct summand of $R$, then $I^2=I$ certainly fails for some non-unital rings. – Batominovski Dec 26 '17 at 14:08

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It would be sufficient to find an integral domain with a nontrivial idempotent ideal, and that we done at this previous question among other places.

rschwieb
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