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Removed the deprecated (abstract-algebra) tag - see the tag info: https://mathoverflow.net/tags/abstract-algebra/info (if there are some other suitable tags, choose them instead.)
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Martin Sleziak
Martin Sleziak
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ideals Ideals and Idempotents in a commutative ring

Let $I$, $J$, and $K$ be pairwasepairwise 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)

ideals and Idempotents in a commutative ring

Let $I$, $J$, and $K$ be pairwase 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)

Ideals and Idempotents in a commutative ring

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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E.R
E.R
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ideals and Idempotents in a commutative ring

Let $I$, $J$, and $K$ be pairwase 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)