We now take a detour from the purely linear-algebraic material to study modules—a generalization of vector spaces where the scalars come from a ring rather than a field. The payoff is immediate: the classification of finitely generated abelian groups reduces to “linear algebra over .”
Recall from Lecture 2 that every abelian group is a -module: the action sends (where for , , and for ). This observation is the bridge between group theory and module theory.
10.1.1 Definitions and Examples
Definition 10.1.1 (Module). Let be a ring (with unity ). A (left) -module is an abelian group equipped with a scalar multiplication , , satisfying:
- for all , .
- for all , .
- for all , .
- for all .
Example 10.1.1 (Vector spaces). If is a field, then a -module is the same thing as a vector space over . So modules generalize vector spaces by allowing the scalars to come from a ring.
Example 10.1.2 (Abelian groups as -modules). Every abelian group is a -module via . Conversely, every -module is an abelian group (just forget the -action, which is determined by the group structure anyway). The categories of abelian groups and -modules are the same.
Example 10.1.3 (The ring as a module over itself). is an -module via left multiplication: . This is a free module of rank 1.
Example 10.1.4 (-modules from ring homomorphisms). If is a ring homomorphism, then any -module becomes an -module via . This is called restriction of scalars.
Example 10.1.5 (Polynomial modules). Let and a vector space over . Any linear operator makes into a -module via . Different operators give different module structures on the same underlying vector space.
10.1.2 Submodules and Ideals
Definition 10.1.2 (Submodule). A submodule of an -module is a subset that is itself an -module under the inherited operations. Equivalently, is a subgroup of that is closed under scalar multiplication: for all , .
Definition 10.1.3 (Ideal). A submodule of (viewed as a module over itself) is called a left ideal. When is commutative (as will be the case for us), left ideals are automatically two-sided ideals: subsets satisfying:
- is an abelian subgroup of .
- and for all .
Example 10.1.6 (Ideals in and ). The ideals of are exactly the subgroups for . Similarly, the ideals of are . In both cases, every ideal is generated by a single element. Rings with this property are called principal ideal domains (PIDs). This is closely related to the Euclidean division algorithm: .
Note. The quotient of an -module by a submodule is again an -module, with . For example, as a -module, and is a -module. The quotient of by an ideal is not just an -module but a ring in its own right.
10.1.3 General Facts about Modules: Nothing Is True
Over a field, every vector space has a basis, every linearly independent set extends to one, and every spanning set contains one. Over a general ring, all of these fail.
Example 10.1.7 (Bases need not exist). The -module (for ) has no basis. Any map sending cannot be injective (since in but in ). So no single-element set is linearly independent, and no basis can exist.
Example 10.1.8 (Linearly independent sets may not be part of a basis). Consider as a -module. The element is linearly independent (if then ). But cannot be extended to a basis of : the only bases of are and , and no set containing is a basis.
Example 10.1.9 (Spanning sets need not contain a basis). The -module is spanned by (since , so ). But is not linearly independent (), and neither subset or spans . So the spanning set contains no basis.
Example 10.1.10 (Submodules of finitely generated modules need not be finitely generated). Let (polynomials in infinitely many variables) and as an -module. Then is generated by . But the ideal (polynomials with zero constant term) is a submodule not finitely generated: any finite subset only involves finitely many variables , so cannot span for large .
By contrast, this pathology does not occur over Noetherian rings (including and ).
10.1.4 Module Homomorphisms
Definition 10.1.4 (Module homomorphism). Let be -modules. A map is a module homomorphism if for all , .
Note. When is commutative, is itself an -module: and . (For noncommutative , this scalar multiplication need not yield an -module homomorphism, so is only an abelian group.) For free modules, things work as expected: (determined by images of basis vectors, just like linear maps between vector spaces). But for non-free modules, can vanish even when both and are nonzero. For instance, when and : any must satisfy , so is killed by in , forcing .
10.1.5 Free Modules and Direct Sums
Definition 10.1.5 (Free module). An -module is free if it has a basis: a set such that every element of can be written uniquely as a finite -linear combination . Equivalently, . The rank of a free module is (well-defined when is commutative).
Note. Over a field, every module (= vector space) is free. Over , free modules are the groups . The module is not free for .
Definition 10.1.6 (Direct sum of modules). The direct sum is the set of pairs with componentwise operations. More generally, consists of tuples with only finitely many nonzero entries. This has the universal property: maps out of correspond to families of maps out of each .
Definition 10.1.7 (Exact sequences of modules). A sequence of -module homomorphisms is exact at if . A short exact sequence means is injective, is surjective, and , i.e., . A short exact sequence splits if . Over a PID, every short exact sequence of finitely generated modules with free splits (analogous to Maschke’s theorem).