Module problems and examples

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 31 January 2012

Module examples

HW:

Let $R$ be a PID. Let $M$ be a finitely generated $R-$module. Show that $M$ is torsion free if and only if $M$ is free.

Example. Let $R=\mathbb{Z}$ and $M=\frac{\mathbb{Z}}{\left(36\right)}\oplus \frac{\mathbb{Z}}{\left(52\right)}.$ Then $M\cong \frac{\mathbb{Z}}{\left(2^2\right)}\oplus \frac{\mathbb{Z}}{\left(3^2\right)}\oplus \frac{\mathbb{Z}}{\left(2^2\right)}\oplus \frac{\mathbb{Z}}{\left(13\right)}.$ Also $M\cong \frac{\mathbb{Z}}{\left(2^2\right)}\oplus \frac{\mathbb{Z}}{\left(2^4\right)}\oplus \frac{\mathbb{Z}}{\left(2^4{3}^2\right)}\oplus \frac{\mathbb{Z}}{\left(2^4{3}^{2}13\right)},$ and this is decomposition given by the decomposition given by the decomposition theorem for finitely generated modules.

Example. Let $R=ℂ\left[t\right]$ and $M=\frac{ℂ\left[t\right]}{ ⟨{\left(t-2\right)}^3{\left(t-5\right)}^4⟩ }.$ Then $M$ is a $ℂ\left[t\right]-$module and hence, it is also a $ℂ-$module. Thus $M$ is a $ℂ-$vector space. There are two natural bases of $M$ as a $ℂ-$vector space $${1,t,t^2,t^3,t^4,t^5,t^6}$$ and since $M$ is isomorphic to $\frac{ℂ\left[t\right]}{ ⟨{\left(t,-,5\right)}^4⟩ }$ we also have natural bases $${\left(1,0\right),\left(t,0\right),\left(t^2,0\right),\left(0,1\right),\left(0,t\right),\left(0,t^2\right),\left(0,t^3\right)}$$ and $${\left(1,0\right),\left(t-2,0\right),\left({\left(t-2\right)}^2,0\right),\left(0,1\right),\left(0,t-5\right),\left(0,{\left(t-5\right)}^2\right),\left(0,{\left(t-5\right)}^3\right)} .$$ The matrix of the action of $t$ with respect to each of these different bases is $$T=\left(\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& -?\\ 1& 0& 0& 0& 0& 0& -?\\ 0& 1& 0& 0& 0& 0& -?\\ 0& 0& 1& 0& 0& 0& -?\\ 0& 0& 0& 1& 0& 0& -?\\ 0& 0& 0& 0& 1& 0& -?\\ 0& 0& 0& 0& 0& 1& -?\end{array}\right)T=\left(\begin{array}{ccccccc}0& 0& -?& 0& 0& 0& 0\\ 1& 0& -?& 0& 0& 0& 0\\ 0& 1& -?& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& -?\\ 0& 0& 0& 1& 0& 0& -?\\ 0& 0& 0& 0& 1& 0& -?\\ 0& 0& 0& 0& 0& 1& -?\end{array}\right)T=\left(\begin{array}{ccccccc}2& 0& 0& 0& 0& 0& 0\\ 1& 2& 0& 0& 0& 0& 0\\ 0& 1& 2& 0& 0& 0& 0\\ 0& 0& 0& 5& 0& 0& 0\\ 0& 0& 0& 1& 5& 0& 0\\ 0& 0& 0& 0& 1& 5& 0\\ 0& 0& 0& 0& 0& 1& 5\end{array}\right).$$

Example. Let $T\in M_6\left(ℂ\right).$ Then let $M={ℂ}^6$ and use $T$ to define an action of $ℂ\left[t\right]$ on $M$. Then $$M\cong \frac{ℂ\left[t\right]}{\left(f_1\right)}\oplus \cdots \oplus \frac{ℂ\left[t\right]}{\left(f_r\right)}$$ where $f_i|f_{i+1}$ and the $f_i$ are monic polynomials with coefficients in $ℂ$.

Problems

1. Let $R$ be an integral domain and let $M$ be an $R-$module. Show that $\mathrm{Tor}M$ is a submodule of $M$.
2. Give an example of a commutative ring $R$ and an $R-$module $M$ such that $\mathrm{Tor}M= {m\in M|m \; is\; not\; linearly\; independent}$ is not an $R-$module.
3. Let $R$ be an integral domain and let $M$ be an $R-$module. Show that $\frac{M}{\left(\mathrm{Tor}M\right)}$ is a torsion free $R-$module.
4. Let $R$ be a PID and let $M$ be a cuclic $R-$module. Show that there exists $a\in R$ such that $M\simeq \frac{R}{Ra}.$
5. Let $R$ be a PID and let $a\in R$. Let ${p_1}^{s_1}{p_2}^{s_2}\cdots {p_k}^{s_k}$ be a factorization of $a$ into a product of irreducible elements such that $p_i$ is not associate to any $p_j$, $i\ne j$. Show that $$\frac{R}{Ra}\simeq \frac{R}{R{p_1}^{s_1}}\oplus \frac{R}{R{p_2}^{s_2}}\oplus \cdots \oplus \frac{R}{R{p_k}^{s_k}}.$$
6. Let $R$ be a PID. Let $a\in R$. Show that $R/Ra$ is a torsion module if $a\ne 0$ and that $R/Ra$ is torsion free if $a=0.$
7. Let $M$ be a finitely generated module over a PID. Show that $M$ is the direct sum of $\mathrm{Tor}M$ and a free module.
8. Let $R$ be a PID and let $M$ be a finitely generated torsion free $R-$module. Show that $M$ is free.

Notes and References

Where are these from?

References

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