Editing Divisible group (section)

==Generalization==

Several distinct definitions generalize divisible groups to divisible modules. The following definitions have been used in the literature to define a '''divisible [[Module (mathematics)|module]]''' ''M'' over a [[Ring (mathematics)|ring]] ''R'':
# ''rM''&thinsp;=&thinsp;''M'' for all nonzero ''r'' in ''R''.{{sfn|Feigelstock|2006}} (It is sometimes required that ''r'' is not a zero-divisor, and some authors{{sfn|Cartan|Eilenberg|1999}} require that ''R'' is a [[Domain (ring theory)|domain]].)
# For every principal left [[Ideal (ring theory)|ideal]] ''Ra'', any [[Module homomorphism|homomorphism]] from ''Ra'' into ''M'' extends to a homomorphism from ''R'' into ''M''.{{sfn|Lam|1999}}{{sfn|Nicholson|Yousif
|2003}} (This type of divisible module is also called ''principally injective module''.)
# For every [[finitely generated module|finitely generated]] left ideal ''L'' of ''R'', any homomorphism from ''L'' into ''M'' extends to a homomorphism from ''R'' into ''M''.{{Citation needed|date=January 2023}}

The last two conditions are "restricted versions" of the [[Baer's criterion]] for [[injective module]]s. Since injective left modules extend homomorphisms from ''all'' left ideals to ''R'', injective modules are clearly divisible in sense 2 and 3.

If ''R'' is additionally a domain then all three definitions coincide.  If ''R'' is a principal left ideal domain, then divisible modules coincide with injective modules.{{sfn|Lam|1999|loc=p.70—73}} Thus in the case of the ring of integers '''Z''', which is a principal ideal domain, a '''Z'''-module (which is exactly an abelian group) is divisible if and only if it is injective.

If ''R'' is a [[Commutative ring|commutative]] domain, then the injective ''R'' modules coincide with the divisible ''R'' modules if and only if ''R'' is a [[Dedekind domain]].{{sfn|Lam|1999|loc=p.70—73}}