Divisible Group - Generalization

Generalization

Several distinct definitions which generalize divisible groups to divisible modules. The following definitions have been used in the literature to define a divisible module M over a ring R:

1. rM=M for all nonzero r in R. (It is sometimes required that r is not a zero-divisor, and some authors require that R is a domain.)
2. For every principal left ideal Ra, any homomorphism from Ra into M extends to a homomorphism from R into M. (This type of divisible module is also called principally injective module.)
3. For every finitely generated left ideal L of R, any homomorphism from L into M extends to a homomorphism from R into M.

The last two conditions are "restricted versions" of the Baer's criterion for injective modules. 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. 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 domain, then the injective R modules coincide with the divisible R modules if and only if R is a Dedekind domain.