Interesting analogy, but it's a little off. The main reason modules are interesting is not as a generalisation of vector spaces, but because they are helpful i

Interesting analogy, but it's a little off. The main reason modules are interesting is not as a generalisation of vector spaces, but because they are helpful in studying rings. Kernels of ring homomorphisms are ideals, which in general are not subrings, but they are modules - and of course every ring is a module over itself. So to study a ring R it pays off to instead study R-modules, since working with them is... you guessed it! Simpler.