Some of my favourite topics in pure mathematics! Two quick general remarks:
I don’t hold such a strong qualitative distinction between the theory of group actions, and in particular linear representations, and the theory of modules. They are both ways to study an object by having it act on auxiliary structures/geometry. Because there are in general fewer tools to study group actions than modules, a lot of pure mathematics is dedicated to linearizing the former to the latter in various ways.
There is another perspective on modules over commutative rings which is central to algebraic geometry: modules are a specific type of sheaves which generalize vector bundles. More precisely, a module over a commutative ring R is equivalent to a “quasicoherent sheaf” on the affine scheme Spec(R), and finitely generated projective modules correspond in this way to vector bundles over Spec(R). Once you internalise this equivalence, most of the basic theory of modules in commutative algebra becomes geometrically intuitive, and this is the basis for many further developments in algebraic geometry.
BTW the geometric perspective might sound abstract (and setting it up rigorously definitely is!) but it is many ways more concrete than the purely algebraic one. For instance, a quasicoherent sheaf is in first approximation a collection of vector spaces (over varying “residue fields”) glued together in a nice way over the topological space Spec(R), and this clarifies a lot how and when questions about modules can be reduced to ordinary linear algebra over fields.
Some of my favourite topics in pure mathematics! Two quick general remarks:
I don’t hold such a strong qualitative distinction between the theory of group actions, and in particular linear representations, and the theory of modules. They are both ways to study an object by having it act on auxiliary structures/geometry. Because there are in general fewer tools to study group actions than modules, a lot of pure mathematics is dedicated to linearizing the former to the latter in various ways.
There is another perspective on modules over commutative rings which is central to algebraic geometry: modules are a specific type of sheaves which generalize vector bundles. More precisely, a module over a commutative ring R is equivalent to a “quasicoherent sheaf” on the affine scheme Spec(R), and finitely generated projective modules correspond in this way to vector bundles over Spec(R). Once you internalise this equivalence, most of the basic theory of modules in commutative algebra becomes geometrically intuitive, and this is the basis for many further developments in algebraic geometry.
BTW the geometric perspective might sound abstract (and setting it up rigorously definitely is!) but it is many ways more concrete than the purely algebraic one. For instance, a quasicoherent sheaf is in first approximation a collection of vector spaces (over varying “residue fields”) glued together in a nice way over the topological space Spec(R), and this clarifies a lot how and when questions about modules can be reduced to ordinary linear algebra over fields.