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

Def Noetherian Ring

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

Prop Every PID is noetherian.

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

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

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

Structure Theorem

Thrm 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.

Def Euclidean Ring

A domain **is called euclidean if it is equipped with a map , called a degree function, such that

1. for all non-zero elements
2. For all **with , there exist **with , where either or .

e.g. with , with , and with , are all euclidean rings.

Prop Every euclidean ring is a PID.

Thrm Structure Theorem for Finitely Generated Abelian Groups

Let be a finitely generated abelian group. There exist and positive integers such that

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