I need a reference which states which of the "normal properties of vector spaces" carry over to free $\mathbb{Z}$-modules.


Especially I am interested in things like: If you have a linear map between two free $\mathbb{Z}$-modules and you choose a basis for its kernel, can you choose a basis of a complementary space so that both together form a basis of the whole space (and the map, viewed only on this complementary space, is an isomorphism on its image)?


Probably this is an easy question for algebra guys.



free Z-modules: Bases etc.
What carries over?


As Peter pointed out, a submodule of a free $\mathbb{Z}$-module though
free need not have a complement. Indeed each submodule of a free
$\mathbb{Z}$-module is free, but a quotient module need not be, for instance
$\mathbb{Z}/2\mathbb{Z}$. Also a $\mathbb{Z}$-module is free if and oly if
it is projective; this entails that a kernel of a map of free modules
does have a complement.


The set $\mathrm{Hom}(F,G)$ for free $\mathbb{Z}$-modules need not be free.
If $F$ is free of countably infinite rank and $G=\mathbb{Z}$, then
$\mathrm{Hom}(F,G)\cong\prod\_{j=1}^\infty\mathbb{Z}$ which remarkably
is not free over $\mathbb{Z}$. But $F\otimes G$ is free for free $F$ and $G$.


