09 - Structure Theorem for Finitely Generated Modules over a Euclidean Domain

Structure Theorem for Finitely Generated Modules over a Euclidean Domain

Let $M$ be a finitely generated module over Euclidean Domain $R$ then

Exists integer $s$ and nonzero non-units $c_1,c_2,\cdots ,c_r\in R$ with $c_1\mid c_2\mid \cdots \mid c_r$ such

$$M\cong \left(\underset{i=1}{\overset{r}{⨁}}R/c_{i}R\right)\oplus R^s$$

Proof

As $R$ is a PID then there is presentation for $M$ with

$$\begin{array}{rl}\psi & :R^m\to R^n\\ \varphi & :R^n\to m\end{array}$$

where $m\le n$ and $\psi$ is injective and $\varphi$ is surjective so

$$M\cong R^n/\mathrm{im}(\psi )$$

If $A$ is the matrix of $\psi$ with respect to standard bases of $R^m$ and $R^n$ then

By Smith Normal Form then
There is normal form for matrices over Euclidean Domain so $A$ can be transformed into
Diagonal Matrix $D$ with diagonal entries $d_1\mid d_2\mid \cdots \mid d_m$ using EROs and ECOs
As EROs and ECOs correspond to changing bases in $R^n$ and $R^m$ then

Exists basis for $R^n$ and $R^m$ with respect to $\psi$ which has matrix $D$
Let $\{f_1,\cdots ,f_n\}$ denote the basis of $R^n$ so image of $\psi$ has basis

$$\{d_1{f}_1,\cdots ,d_m{f}_m\}$$

Define map

$$\theta :R^n\to \left(\underset{i=1}{\overset{m}{⨁}}R/d_{i}R\right)\oplus R^n$$

For $m={\sum }_{i=1}^n{a}_i{f}_i\in M$ then

$$\theta \left(\sum _{i=1}^n{a}_i{f}_i\right)i$$

where $\theta$ is surjective and $\ker (\varphi )$ is submodule generated by

$$\{d_i{f}_i:1\le i\le m\}=(a_1+d_{1}R,\cdots ,a_m+d_{m}R,a_{m+1},\cdots ,a_n)$$

which is $\mathrm{im}(\psi )$

By First Isomorphism Theorem then

$$M\cong R^k/\mathrm{im}(\psi )\cong \left(\underset{i=1}{\overset{m}{⨁}}R/d_{i}R\right)\oplus R^n$$

As $\psi$ is injective then $d_i$ are all nonzero
If $d_i$ is a unit then $R/d_{i}R=0$