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.
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.