## Modules

Last update: 02 February 2012

## Modules

Let $R$ be a ring.

• An abelian group is a set $M$ with an operation $+:M×M\to M$ such that
1. If ${m}_{1},{m}_{2},{m}_{3} \in M$ then $\left({m}_{1}+{m}_{2}\right)+{m}_{3}={m}_{1}+\left({m}_{2}+{m}_{3}\right),$
2. If ${m}_{1},{m}_{2} \in M$ then ${m}_{1}+{m}_{2}={m}_{2}+{m}_{1},$
3. There is an element $0\in M$ such that $0+m=m+0,$ for all $m\in M,$
4. If $m\in M$ there is an element $-m\in M$ such that $m+\left(-m\right)=\left(-m\right)+m=0.$
• An $R-$module is an abelian group $M$ with an $R-$action $\cdot :R×M\to M$ such that
1. If $r,{r}_{1},{r}_{2},\in ,R$ and $m,{m}_{1},{m}_{2},\in ,M$ then $\left({r}_{1}+{r}_{2}\right)m={r}_{1}m+{r}_{2}m$ and $r\left({m}_{1}+{m}_{2}\right)=r{m}_{1}+r{m}_{2},$
2. If $m\in M$ then $1\cdot m=m$,
3. If ${r}_{1},{r}_{2} \in R$ and $m\in M$ then $\left({r}_{1}{r}_{2}\right)m={r}_{1}\left({r}_{2}m\right).$
• The regular $R-$module is the abelian group $R$ with $R-$action given by left multiplication. Equivalently, the regular $R-$module is the set and $R-$action $R×\stackrel{\to }{R}\to \stackrel{\to }{R}$ given by

HW: Show that the regular $R-$module is an $R-$module.

Examples.

1. $ℤ/kℤ$ is a $ℤ-$module.
2. If $R={M}_{n}\left(ℂ\right)$ and $V={M}_{n×1}\left(ℂ\right)$ then $V$ is an $R-$module.
3. Let $R$ be a ring. Then $R$ is an $R-$module.
4. If $R$ is a ring and $I$ is an ideal of $R$ then $R/I$ is an $R-$module.

Let $𝔽$ be a field.

1. The category of abelian groups is equivalent to the category of $ℤ-$modules.
2. The category of vector spaces over $𝔽$ is equivalent to the category of $𝔽-$modules.
3. The category of vector spaces $V$ over $𝔽$ with a linear transformation $T:V\to V$ is equivalent to the category of $𝔽\left[x\right]-$modules.

Let $R$ be a ring. Let $M$ be an $R-$module.

• A submodule of $M$ is a subset $N\subseteq M$ such that
1. If ${n}_{1},{n}_{2} \in N$ then ${n}_{1}+{n}_{2}\in N;$
2. $0\in N;$
3. If $n\in N$ then $-n\in N;$
4. If $r\in R$ and $n\in N$ then $rn\in N.$
• A left ideal is a submodule of the regular $R-$module.

HW: Show that 0 is a submodule of $M$.

HW: Show that $M$ is a submodule of $M$.

HW: Show that $L$ is a left ideal of $R$ if and only if $L$ is a subset of $R$ such that

1. If ${l}_{1},{l}_{2} \in L$ then ${l}_{1}+{l}_{2}\in L,$
2. $0\in L,$
3. If $l\in L$ then $-l\in L,$
4. If $r\in R$ and $l\in L$ then $rl\in L.$

Let $R$ be a ring and let $M$ be an $R-$module.

• The annihilator of an element $m\in M$ is $\mathrm{ann}\left(m\right)= {r\in R|rm=0} .$
• The annihilator of $M$ is the set

HW: Show that $\mathrm{ann}\left(m\right)$ is a left ideal of $R$.

HW: Show that $\mathrm{span}\left(m\right)\cong R/\mathrm{ann}\left(m\right).$

HW: Show that $\mathrm{ann}\left(M\right)$ is an ideal of $R.$

HW: Give an example of a ring such that $\mathrm{ann}\left(R\right)=0.$

HW: Give an example of a ring such that $\mathrm{ann}\left(R\right)\ne 0.$

HW: Give an example when $\mathrm{ann}\left(m\right)$ is not $0$ and not $R$.

Let $R$ be a ring.

• A simple module is an $R-$module $M$ which has no submodules except $0$ and $M$.
• An indecomposable module is an $R-$module $M$ that cannot be written as $M=N\oplus P$ for any two submodules $N,P$ of $M$.
• A cyclic module is an $R-$module $M$ that is generated by a single element.

HW: Give an example of a simple module.

HW: Give an example of a ring such that $R$ is a simple $R-$module.

Let $R$ be a ring and let $M$ and $N$ be $R-$modules.

• An $R-$module homomorphism is a function $f:M\to N$ such that
1. If ${m}_{1},{m}_{2} \in M$ then $f\left({m}_{1}+{m}_{2}\right)=f\left({m}_{1}\right)+f\left({m}_{2}\right),$
2. If $r\in R$ and $m\in M$ then $f\left(rm\right)=rf\left(m\right).$
• An endomorphism of $M$ is a homomorphism $f:M\to M.$

Let $M$ and $N$ be $R-$modules.

1. The set ${\mathrm{Hom}}_{R}\left(M,N\right)$ with addition and $R-$action given by is an $R-$module.
2. The abelian group ${\mathrm{End}}_{R}\left(M\right)$ with multiplication given by is a ring.

Let $R$ be a ring.

1. If $M$ is a simple $R-$module then ${\mathrm{End}}_{R}\left(M\right)$ is a division ring.
2. ${\mathrm{End}}_{R}\left(R\right)\cong {R}^{\mathrm{op}}.$

## Notes and References

Where are these from?

References?