MATH2401 Abstract Algebra (3): Module

This article replaces vector spaces by modules and follows the resulting path from presentations and Smith normal form to finitely generated abelian groups and rational canonical form.

Collections of MATH2401 Abstract Algebra notes:

1. Group;
2. Ring;
3. Module (this article);
4. Field;
5. Action;
6. Galois Theory.

Replacing a field by a ring removes division by nonzero scalars. Modules need not have bases, and submodules of free modules need not be direct summands. Throughout this article is a nonzero ring with identity.

Module basics

Definition. A left -module is an abelian group with an action satisfying

A submodule is an additive subgroup closed under the action of . The submodules of the left regular module are exactly the left ideals of .

Definition. An -linear map preserves addition and satisfies . Its kernel and image are submodules. Its cokernel is .

In particular, every -linear map is a group homomorphism between the underlying additive groups of its modules.

Definition. For a submodule , the quotient has

Products and direct sums of modules use componentwise operations.

Theorem. First isomorphism theorem for modules. For an -linear map ,

Linear operators as modules

Let be a field. If is an -vector space and , define

Conversely, restricting an -module to the constant polynomials gives an -vector space and multiplication by gives an -linear operator.

Proposition. The constructions above give an equivalence between -modules and pairs consisting of an -vector space and an -linear endomorphism. Moreover, -linear maps are exactly the -linear maps commuting with the operators.

Free modules and presentations

Definition. A subset is a basis if every has a unique finite expression . A module with a basis is free. If , then .

For free modules with chosen finite bases, homomorphisms are represented by matrices (acting on column vectors). Over a noncommutative ring this convention is for left modules and coefficients must be placed consistently.

Proposition. Every module generated by is a quotient of . More precisely, the map is surjective, and its kernel is the module of relations among the generators.

Finite generation does not imply freeness: is generated by one element as a -module but is not free. A nonzero relation in one chosen presentation does not itself prove nonfreeness. For example,

this presentation has a nonzero relation module although its quotient is free.

For nonzero commutative rings, any two bases of a finite free module have the same cardinality; this number is its rank. (Arbitrary noncommutative rings need not have this invariant-basis-number property.)

Smith normal form

Theorem. Smith normal form. Let be a PID and . There are invertible matrices and such that

where every . The nonzero , normalized up to units, are uniquely determined and are called the invariant factors.

Allowed reductions. Smith reduction uses the following invertible operations, applied to either rows or columns:

1. interchange two rows (or two columns);

2. multiply a row (or column) by a unit of ;

3. add a multiple of one row (or column) to another.

Equivalently, these operations replace by with and . ◻

Corollary. Every subgroup of a free abelian group is free of rank at most .

Proof. Induct on , with immediate. Choose a coordinate projection . If , apply induction in . Otherwise for some . Put and choose with . For , write ; then , uniquely. Hence

Induction makes free of rank at most , completing the proof. ◻

Proposition. Every finitely generated module over a PID has a finite presentation

equivalently, is the cokernel of an injective map between finite free modules.

Proof. For a surjection and an isomorphism , the presentation is

 ◻

Finitely generated abelian groups

Theorem. Structure theorem for finitely generated abelian groups. Every finitely generated abelian group has a decomposition

The rank and invariant factors are unique.

Uniqueness. The free rank is intrinsic:

The torsion subgroup and its -primary subgroup are also intrinsic. If , then for every ,

These dimensions recover the multiset of exponents for every prime . Moreover, . Padding the exponent lists on the left with zeros to length and using uniquely recovers each . ◻

The first summand is the free part and the finite direct sum is the torsion part. The torsion part can alternatively be decomposed into cyclic groups of prime-power order (the elementary-divisor form). Its exponent is when , i.e. the least positive integer annihilating every torsion element.

Application to linear operators

Since is a PID, the structure theorem applies to the -module associated with a linear operator .

Theorem. Invariant-factor decomposition. Let be finite-dimensional over and let . There are unique monic nonconstant polynomials such that, as -modules,

There is no free summand, because is infinite-dimensional over . The companion matrices of the give the rational canonical form of .