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Let $R$ be a Prufer domain. If $0 \ne a \in R$ is such that $Ra \cap Rb$ is principal ideal for every $b \in R$, then is it true that $Ra+Rb$ is also principal for every $b\in R$ ?

Might be related On GCD and LCM of elements in integral domain with Krull-dimension 1

Over Prufer domains, torsion-free modules are flat , so if $K$ is the fraction field of $R$ then any subring $S$ of $K$ containing $R$, is a flat $R$-module, hence $S$ over $R$ has Going Down property . So this On GCD and LCM of elements in integral domains which has the property that any over ring has Going Down is related.

Let $R$ be a Prufer domain. If $0 \ne a \in R$ is such that $Ra \cap Rb$ is principal ideal for every $b \in R$, then is it true that $Ra+Rb$ is also principal for every $b\in R$ ?

Might be related On GCD and LCM of elements in integral domain with Krull-dimension 1

Over Prufer domains, torsion-free modules are flat , so if $K$ is the fraction field of $R$ then any subring $S$ of $K$ containing $R$, is a flat $R$-module, hence $S$ over $R$ has Going Down property . So this On GCD and LCM of elements in integral domains which has the property that any over ring has Going Down is related.

Let $R$ be a Prufer domain. If $0 \ne a \in R$ is such that $Ra \cap Rb$ is principal ideal for every $b \in R$, then is it true that $Ra+Rb$ is also principal for every $b\in R$ ?

Over Prufer domains, torsion-free modules are flat , so if $K$ is the fraction field of $R$ then any subring $S$ of $K$ containing $R$, is a flat $R$-module, hence $S$ over $R$ has Going Down property . So this On GCD and LCM of elements in integral domains which has the property that any over ring has Going Down is related.

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Let $R$ be a Prufer domain. If $0 \ne a \in R$ is such that $Ra \cap Rb$ is principal ideal for every $b \in R$, then is it true that $Ra+Rb$ is also principal for every $b\in R$ ?

Might be related On GCD and LCM of elements in integral domain with Krull-dimension 1

Over Prufer domains, torsion-free modules are flat , so if $K$ is the fracyionfraction field of $R$ then any subring $S$ of $K$ containing $R$, is a flat $R$-module, hence $S$ over $R$ has Going Down property . So this On GCD and LCM of elements in integral domains which has the property that any over ring has Going Down is related.

Let $R$ be a Prufer domain. If $0 \ne a \in R$ is such that $Ra \cap Rb$ is principal ideal for every $b \in R$, then is it true that $Ra+Rb$ is also principal for every $b\in R$ ?

Might be related On GCD and LCM of elements in integral domain with Krull-dimension 1

Over Prufer domains, torsion-free modules are flat , so if $K$ is the fracyion field of $R$ then any subring $S$ of $K$ containing $R$, is a flat $R$-module, hence $S$ over $R$ has Going Down property . So this On GCD and LCM of elements in integral domains which has the property that any over ring has Going Down is related.

Let $R$ be a Prufer domain. If $0 \ne a \in R$ is such that $Ra \cap Rb$ is principal ideal for every $b \in R$, then is it true that $Ra+Rb$ is also principal for every $b\in R$ ?

Might be related On GCD and LCM of elements in integral domain with Krull-dimension 1

Over Prufer domains, torsion-free modules are flat , so if $K$ is the fraction field of $R$ then any subring $S$ of $K$ containing $R$, is a flat $R$-module, hence $S$ over $R$ has Going Down property . So this On GCD and LCM of elements in integral domains which has the property that any over ring has Going Down is related.

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user111524
user111524

GCD and LCM of elements in Prufer domain

Let $R$ be a Prufer domain. If $0 \ne a \in R$ is such that $Ra \cap Rb$ is principal ideal for every $b \in R$, then is it true that $Ra+Rb$ is also principal for every $b\in R$ ?

Might be related On GCD and LCM of elements in integral domain with Krull-dimension 1

Over Prufer domains, torsion-free modules are flat , so if $K$ is the fracyion field of $R$ then any subring $S$ of $K$ containing $R$, is a flat $R$-module, hence $S$ over $R$ has Going Down property . So this On GCD and LCM of elements in integral domains which has the property that any over ring has Going Down is related.