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.