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Is the category of rings locally smallco-well-powered?

Dear colleagues,

Can anybody explain me if a category of (associative) rings is co-well-powered (this is the MacLane definition, in Russian literature this is called "locally small from the right side")? I mean, for any ring Ait is well-powered, of course, since for any ring A one can easily find a skeleton in the category Mono(A) (of all monomorphisms with values in A) and this will be a set (the set of all subrings in A), but . But is it true, that it is co-well-powered, i.e. for any ring A there exists a skeleton in the category Epi(A) (of all epimorphisms from A into other rings) and is this skeleton again a set?

Thank you in advance, Sergei Akbarov
show/hide this revision's text 1

Is the category of rings locally small?

Dear colleagues,

Can anybody explain me if a category of (associative) rings is locally small? I mean, for any ring A, of course, one can easily find a skeleton in the category Mono(A) (of all monomorphisms with values in A) and this will be a set (the set of all subrings in A), but is it true, that there exists a skeleton in the category Epi(A) (of all epimorphisms from A into other rings) and is this skeleton again a set?

Thank you in advance, Sergei Akbarov