Modules

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

Last update: 02 February 2012

Let $R$ be a ring.

An abelian group is a set M with an operation +:M×M→M such that
1. If $m_{1}, m_{2}, m_{3} \in M$ then $(m_{1} + m_{2}) + m_{3} = m_{1} + (m_{2} + m_{3}),$
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 $m + (- m) = (- m) + m = 0 .$
An $R -$ module is an abelian group M with an R-action ⋅:R×M→M such that
1. If $r r_{1} r_{2} \in R$ and $m m_{1} m_{2} \in M$ then $(r_{1} + r_{2}) m = r_{1} m + r_{2} m$ and $r (m_{1} + m_{2}) = r m_{1} + r m_{2},$
2. If $m \in M$ then $1 \cdot m = m$ ,
3. If $r_{1}, r_{2} \in R$ and $m \in M$ then $(r_{1} r_{2}) m = r_{1} (r_{2} m) .$
The regular $R -$ module is the abelian group $R$ with $R -$ action given by left multiplication. Equivalently, the regular $R -$ module is the set $\vec{R} = {\vec{r}| r \in R} with addition given by \vec{r_{1}} + \vec{r_{2}} = \vec{r_{1} + r_{2}}, for all \vec{r_{1}}, \vec{r_{2}} \in \vec{R},$ and $R -$ action $R \times \vec{R} \to \vec{R}$ given by $r \vec{r_{1}} = \vec{r r_{1}}, for all r \in R and all \vec{r_{1}} \in \vec{R} .$

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

Examples.

Let $𝔽$ be a field.

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

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

A submodule of M is a subset N⊆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 $r n \in N .$
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

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

The annihilator of an element $m \in M$ is $ann (m) = {r \in R| r m = 0} .$
The annihilator of $M$ is the set $ann (M) = {r \in R| r m = 0 for all m \in M} .$

HW: Show that $ann (m)$ is a left ideal of $R$ .

HW: Show that $span (m) ≅ R / ann (m) .$

HW: Show that $ann (M)$ is an ideal of $R .$

HW: Give an example of a ring such that $ann (R) = 0 .$

HW: Give an example of a ring such that $ann (R) \neq 0 .$

HW: Give an example when $ann (m)$ 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→N such that
1. If $m_{1}, m_{2} \in M$ then $f (m_{1} + m_{2}) = f (m_{1}) + f (m_{2}),$
2. If $r \in R$ and $m \in M$ then $f (r m) = r f (m) .$
An endomorphism of $M$ is a homomorphism $f : M \to M .$ $\begin{array}{l} {Hom}_{R} (M, N) = {f : M \to N| f is an R -module homomorphism} \\ {End}_{R} (M) = {Hom}_{R} (M, M) \end{array}$

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

The set ${Hom}_{R} (M, N)$ with addition and $R -$ action given by $(f_{1} + f_{2}) (m) = f_{1} (m) + f_{2} (m), and (r f) (m) = f (r m), for all r \in R, m \in M,$ is an $R -$ module.
The abelian group ${End}_{R} (M)$ with multiplication given by $(f_{1} f_{2}) (m) = f_{1} (f_{2} (m)), for all m \in M,$ is a ring.

Let $R$ be a ring.

Notes and References

Where are these from?

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