## Module problems and examples

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 $R=ℤ$ and $M=\frac{ℤ}{\left(36\right)}\oplus \frac{ℤ}{\left(52\right)}.$ Then $M\cong \frac{ℤ}{\left({2}^{2}\right)}\oplus \frac{ℤ}{\left({3}^{2}\right)}\oplus \frac{ℤ}{\left({2}^{2}\right)}\oplus \frac{ℤ}{\left(13\right)}.$ Also $M\cong \frac{ℤ}{\left({2}^{2}\right)}\oplus \frac{ℤ}{\left({2}^{4}\right)}\oplus \frac{ℤ}{\left({2}^{4}{3}^{2}\right)}\oplus \frac{ℤ}{\left({2}^{4}{3}^{2}13\right)},$ 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 $1 t t2 t3 t4 t5 t6$ and since $M$ is isomorphic to $\frac{ℂ\left[t\right]}{ ⟨{\left(t,-,5\right)}^{4}⟩ }$ we also have natural bases $10 t0 t20 01 0t 0t2 0t3$ and $10 t-20 t-22 0 01 0t-5 0 t-52 0 t-53 .$ The matrix of the action of $t$ with respect to each of these different bases is $T= 0 0 0 0 0 0 -? 1 0 0 0 0 0 -? 0 1 0 0 0 0 -? 0 0 1 0 0 0 -? 0 0 0 1 0 0 -? 0 0 0 0 1 0 -? 0 0 0 0 0 1 -? T= 0 0 -? 0 0 0 0 1 0 -? 0 0 0 0 0 1 -? 0 0 0 0 0 0 0 0 0 0 -? 0 0 0 1 0 0 -? 0 0 0 0 1 0 -? 0 0 0 0 0 1 -? T= 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 5 0 0 0 0 0 0 1 5 0 0 0 0 0 0 1 5 0 0 0 0 0 0 1 5 .$

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 $M≅ ℂ[t] (f1) ⊕⋯⊕ ℂ[t] (fr)$ 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 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 $R Ra ≃ R R p1s1 ⊕ R R p2s2 ⊕ ⋯ ⊕ R R pksk .$
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?