Let $\mathcal{C}$ be the category of modules over a ring. Let also $\mathcal{F}$ be a class of objects in an abelian category closed under pure subobject (pure quotient) and direct limit. Is $\mathcal{F}$ closed under pure quotient (pure subobject)?

What can be said about the category of quasi-coherent sheaves?

Let $\mathcal{C}$ be the category of modules over a ring. Let also $\mathcal{F}$ be a class of objects in $\mathcal C$ closed under pure subobject (pure quotient) and direct limit. Is $\mathcal{F}$ closed under pure quotient (pure subobject)?

What can be said about the category of quasi-coherent sheaves?

Let $\mathcal{F}$ be a class of objects in an abelian category closed under pure subobject (pure quotient) and direct limit. Is $\mathcal{F}$ closed under pure quotient (pure subobject)?

Let $\mathcal{C}$ be the category of modules over a ring. Let also $\mathcal{F}$ be a class of objects in an abelian category closed under pure subobject (pure quotient) and direct limit. Is $\mathcal{F}$ closed under pure quotient (pure subobject)?

What can be said about the category of quasi-coherent sheaves?