free Z-modules: Bases etc. 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.
 A: 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 $G$.
A: You can write your map as a matrix. Moreover, you can choose a different basis so that the matrix is in the Smith's normal formal. The complement to the kernel exists only if the Smith's normal form contains only ones and zeroes. That is about it and should be explained in many Algebra books, say, Artin's Algebra.
A: There is no complementary space in general: Consider the multiplication by 2 map from $\mathbb{Z}$ to itself ...
