Presentation of Finitely Generated Modules

Diagonalising Integral Matrices

Thrm Smith Normal Form

Let . There exists invertible matrices and such that is diagonal with positive entries on the diagonal that with . This form is unique.

Presentation of Finitely Generated Modules

Def Presentations of a Finitely Generated Module

Let **be a ring and **a map of finitely generated -modules. Then is a submodule of **and we can form the quotient . In particular, if is a matrix, it defines a map of free modules , **and we can look at the quotient . We say that **is presented **by the matrix .

e.g. gives

Thrm Let be in . Then the following matrices present the same quotient:

• in Smith Normal Form with and
• obtained from by deleting a column of zeros
• obtained from by removing the th row and th column if the th column is the standard basis element

Proof Suppose we have and is in the Smith Normal Form. Then

as and are invertible, thus preserve the module.

Corollary Any presentation of a finitely generated module in the form of can be written as

Thrm Suppose is a principle integral domain(PID), then for all finitely generated -module , there exists such that

Noetherian Rings

Noetherian Ring

A ring is called noetherian if every ideal is finitely generated.

e.g. Clearly, every PID is noetherian.

Proposition

Let be a ring, and be -modules. Then if and are finitely generated so is .

Proposition

Let be a noetherian ring. Every submodule of a finitely generated -module is finitely generated.

Proposition

Let be a PID. A submodule of a finitely generated free module over a is free and .

Structure Theorem

Structure of Finitely Generated Modules over PID

Let be a PID and a finitely generated -module. There exist and all non-zero in such that

Moreover, and the ideals are uniquely determined.

The Structure Theorem for Finitely Generated Abelian Groups

Every finite abelian group is isomorphic to a direct product (sum) of cyclic groups:
such that divides for all , and .
More generally, every finitely generated abelian group is isomorphic to

where is called the rank of , divides for all , and is called the torsion subgroup of .

Corollary Structure of Linear Operators on Finite Dimensional Vector Spaces

Let be an operator on an -dimensional space over a field . There exist monic polynomials uniquely determined such that

is a -invariant decomposition with cyclic with the characteristic polynomial of