Let $x,y$ be nonzero elements in a commutative ring $R$. Is $(x)\cap (y)$ always finitely generated? What if we further assume that $R$ is an integral domain? Can we construct non-Noetherian non-local semi-local integral domains whose every maximal ideal is principal (in the local case, there are certain valuation rings.)?
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1$\begingroup$ To your first question about $R$ being a domain and $(x)\cap (y)$ not being finitely generated, see math.stackexchange.com/q/296653 $\endgroup$– Uriya FirstCommented Apr 23, 2020 at 7:36
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$\begingroup$ Excellent, thanks! $\endgroup$– sagnik chakrabortyCommented Apr 24, 2020 at 13:02
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1$\begingroup$ One way to force finite generation of $(x)\cap (y)$ is to make this ideal equal to $(xy)$ , and one nice case where this happens is that $x$ is not a zero divisor on $R/(y)$ $\endgroup$– sdeyCommented Jun 17, 2020 at 2:15
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