It's all a generalization of Linear Algebra!
: The study of vector spaces over fields.
  , 
   , ,    
: The study of modules over commutative rings.
Vector spaces are modules over fields (the field doesn't have to be
, it could be
Abelian groups are modules over the ring of integers,
There are many other rings and modules.
We know how to decompose a vector space into products and how to give generators for a vector space. What about modules?