Editing Dedekind-infinite set (section)

==Generalizations==
Expressed in [[category theory|category-theoretical]] terms, a set ''A'' is Dedekind-finite if in the [[category of sets]], every [[monomorphism]] {{nowrap|''f'' : ''A'' → ''A''}} is an [[Isomorphism#Category_theoretic_view|isomorphism]]. A [[von Neumann regular ring]] ''R'' has the analogous property in the category of (left or right) ''R''-[[module (mathematics)|modules]] if and only if in ''R'', {{nowrap|1=''xy'' = 1}} implies {{nowrap|1=''yx'' = 1}}. More generally, a ''[[Dedekind-finite ring]]'' is any ring that satisfies the latter condition. Beware that a ring may be Dedekind-finite even if its underlying set is Dedekind-infinite, e.g. the [[Integer#Algebraic_properties|integers]].