Let $I$, $J$, and $K$ be pairwise comaximal ideals of a commutative ring $R$ with $1$ with the property that if $x^2-x\in I$, then there exists $e^2=e\in J$ such that $x-e\in I$ and if $y^2-y\in K$, then there exists $f^2=f\in J$ such that $y-f\in K$, where $x, y\in R$. I am looking for conditions on $I$, $J$, and $K$ under which we can find an idempotent element $h^2=h\in R$ such that $h\in I$ and $1-h\in K$.( Perhaps such an $h$ This can be found without any assumption)
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1$\begingroup$ Note: $I,J$ comaximal means $I+J=R$. $\endgroup$– YCorCommented Jul 16, 2018 at 18:52
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2$\begingroup$ What's your favorite example of such a ring and such a trio of ideals? $\endgroup$– Neil EpsteinCommented Jul 19, 2018 at 16:45
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