## Modules

Let $R$ be a ring with identity $1\in R$.

• A left $R$-module is a set $M$ with functions  $\begin{array}{ccc}M×M& \to & M\\ \left({m}_{1},{m}_{2}\right)& ⟼& {m}_{1}+{m}_{2}\end{array}$     and     $\begin{array}{ccc}R×M& \to & M\\ \left(r,m\right)& ⟼& rm\end{array}$,
addition and $R$-action, such that
(a)   If ${m}_{1},{m}_{2},{m}_{3}\in M$ then $\left({m}_{1}+{m}_{2}\right)+{m}_{3}=\left({m}_{1}+\left({m}_{2}+{m}_{3}\right)$,
(b)   If ${m}_{1},{m}_{2}\in M$ then ${m}_{1}+{m}_{2}={m}_{2}+{m}_{1}$,
(c)   There exists a zero, $0\in M$, such that if $m\in M$ then $0+v=v+0=v$,
(d)   If $m\in M$ then there exists an additive inverse to $m$, $-m\in M$, such that $\left(-m\right)+m=m+\left(-m\right)=0$,
(e)   If ${r}_{1},{r}_{2}\in R$ and $m\in M$ then ${r}_{1}\left({r}_{2}m\right)=\left({r}_{1}{r}_{2}\right)m$,
(f)   If $m\in M$ then $1m=m$,
(g)   If $r\in R$ and ${m}_{1},{m}_{2}\in M$ then $r\left({m}_{1}+{m}_{2}\right)=r{m}_{1}+r{m}_{2}$,
(h)   If ${r}_{1},{r}_{2}\in R$ and $m\in M$ then $\left({r}_{1}+{r}_{2}\right)m={r}_{1}m+{r}_{2}m$,

$R$-modules are the analogues of group actions except for rings.

Note that conditions (a), (b), (c) and (d) in the definition of a left $R$-module imply that every left $R$-module is an abelian group under addition.

HW: Show, using Ex. 2.2.5, Part I, that the element $0\in M$ is unique.
HW: Show, using Ex. 2.2.5, Part I, that if $m\in M$ then the element $-m\in M$ is unique.
HW: Show that if $M$ is a left $R$-module and $m\in M$ then $0\cdot m=0$.

Important examples of modules are:

(a)   If $R$ is a ring then $R$ is a left $R$-module.
(b)   All abelian groups are $ℤ$-modules.
(c)   If $R$ is a field then the $R$-modules are vector spaces.

$R$-module homomorphisms are for comparing $R$-modules.

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

• An $R$-module homomorphism from $M$ to $N$ is a function $f:M\to N$ such that
(a)   If ${m}_{1},{m}_{2}\in M$ then $f\left({m}_{1}+{m}_{2}\right)=f\left({m}_{1}\right)+f\left({m}_{2}\right)$,
(b)   If $r\in R$ and $m\in M$ then $f\left(rm\right)=rf\left(m\right)$.
• A $R$-module isomorphism is a bijective $R$-module homomorphism.
• Two left $R$-modules $M$ and $N$ are isomorphic, $M\simeq N$, if there exists a vector space isomorphism $f:M\to N$ between them.

Note that condition (a) in the definition of an $R$-module homomorphism implies that $f$ is a group homomorphism.
HW: Show that if $M$ and $N$ are left $R$-modules and if $f:M\to N$ is an $R$-module homomorphism then $f\left({0}_{M}\right)={0}_{N}$, where ${0}_{M}$ and ${0}_{N}$ are the zeros in $M$ and $N$, respectively.
HW: Let $f:M\to N$ be an $R$-module homomorphism. Show that if $m\in M$ then $f\left(-m\right)=-f\left(m\right)$.

• A submodule of an $R$-module $M$ is a subset $M\subseteq M$ such that
(a)   If ${n}_{1},{n}_{2}\in N$ then ${n}_{1}+{n}_{2}\in N$,
(b)   $0\in N$,
(c)   If $n\in N$ then $-n\in N$,
(d)   If $n\in N$ and $r\in R$ then $rn\in N$.
• The zero $R$-module, $\left(0\right)$, is the set containing only $0$ with operations $0+0=0$ and $r\cdot 0=0$, for $r\in R$.

• Let $M$ be a left $R$-module and let $S$ be a subset of $M$. The submodule generated by $S$ is the submodule $\left(S\right)$ of $M$ such that
(a)   $S\subseteq \left(S\right)$,
(b)   If $T$ is a submodule of $M$ and $S\subseteq T$ then $\left(S\right)\subseteq T$.

The submodule $\left(S\right)$ is the smallest submodule of $M$ containing $S$. Think of $\left(S\right)$ as gotten by adding to $S$ exactly those elements of $V$ that are needed to make a submodule.

#### Cosets

• A subgroup of a left $R$-module $M$ is a subset $N\subseteq M$ such that
(a)   If ${n}_{1},{n}_{2}\in N$ then ${n}_{1}+{n}_{2}\in N$,
(b)   $0\in N$,
(c)   If $n\in N$ then $-n\in N$,

Let $M$ be a left $R$-module and let $N$ be a subgroup of $M$. We will use the subgroup $N$ to divide up the module $M$.

• A coset of $N$ in $M$ is a set $m+N=\left\{m+n\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}n\in N\right\}$, where $m\in M$.
• $M/N$ (pronounced "$M$ mod $N$") is the set of cosets of $N$ in $M$.

Let $M$ be a left $R$-module and let $N$ be a subgroup of $M$. Then the cosets of $N$ in $M$ partition $M$.

Notice that the proofs of Proposition (mdptn) and Proposition (gpptn) are essentially the same.
HW: Write a very short proof of Proposition (mdptn) by using (gpptn).

#### Quotient modules $↔$ Submodules

Let $M$ be a left $R$-module and let $N$ be a subgroup of $M$. We can try to make the set $M/N$ of cosets of $N$ in $M$ into an $R$-module by defining an addition operation and an action of $R$. This doesn't work with just any subgroup of $N$, the subgroup must be a submodule.

Let $N$ be a subgroup of a left $R$-module $M$. Then $N$ is a submodule of $M$ if and only if $M/N$ with operations given by $(m1+N) + (m2+N) = (m1+m2) +N and r (m+N) = r m+N$ is a left $R$-module.

Notice that the proofs of Proposition 2.2.5 and Proposition 1.1.8 are essentially the same.
HW: Write a shorter proof of Proposition 2.2.5 by using Proposition 1.1.8.

• The quotient module $M/N$ is the left $R$-module of cosets of a submodule $N$ of an $R$-module vector $M$ with operations given by $\left({m}_{1}+N\right)+\left({m}_{2}+N\right)=\left({m}_{1}+{m}_{2}\right)+N$ and $r\left(m+N\right)=rm+N$.

We have made $M/N$ into a left $R$-module when $N$ is a submodule of $V$.

HW: Show that if $N=M$ then $M/N\simeq \left(0\right)$.

#### Kernel and image of a homomorphism

• The kernel of an $R$-module homomorphism $f:M\to N$ is the set $kerf={m∈M | f(m)=0N},$ where ${0}_{N}$ is the zero element of $N$.
• The image of an $R$-module homomorphism $f:M\to N$ is the set $imf={f(m) | m∈M }.$

Let $f:M\to N$ be a linear transformation. Then

(a)   $\mathrm{ker}f$ is a submodule $M$.
(b)   $\mathrm{im}f$ is a submodule of $N$.

Let $f:M\to N$ be a linear transformation. Let ${0}_{M}$ be the zero element of $M$. Then

(a)   $\mathrm{ker}f=\left\{{0}_{M}\right\}$ if and only if $f$ is injective.
(b)   $\mathrm{im}f=N$ if and only if $f$ is surjective.

Notice that the proof of Proposition (mdinjsur)(b) does not use the fact that $f:M\to N$ is a homomorphism, only the fact that $f:M\to N$ is a function.

(a)   Let $f:M\to N$ be an $R$-module homomorphism and let $K=\mathrm{ker}f$. Define $f^: M/kerf ⟶ N m+K ⟼ f(m)$ Then $\stackrel{^}{f}$ is a well defined injective $R$-module homomorphism.
(b)   Let $f:M\to N$ be an $R$-module homomorphism and define $f′: M ⟶ imf m ⟼ f(m)$ Then $f\prime$ is a well defined surjective $R$-module homomorphism.
(c)   If $f:M\to N$ is an $R$-module homomorphism then $M/kerf≃imf,$ where the isomorphism is an $R$-module isomorphism.

#### Direct sums

Suppose $M$ and $N$ are $R$-modules. The idea is to make $M×N$ into an $R$-module.

• The direct sum $V\oplus W$ of two left $R$-modules $M$ and $N$ is the set $M×N$ with operations given by $(m1,n1) + (m2,n2) = (m1+m2, n1+n2) and r(m,n)= (rm,rn) ,$ for $m,{m}_{1},{m}_{2}\in M$, $n,{n}_{1},{n}_{2}\in N$ and $r\in R$. The operations in $M\oplus N$ are componentwise.
• More generally, given left $R$-modules ${M}_{1},{M}_{2},\dots ,{M}_{s}$, the direct sum ${M}_{1}\oplus {M}_{2}\oplus \cdots \oplus {M}_{s}$ is the set ${M}_{1}×{M}_{2}×\cdots ×{M}_{s}$ with the operations given by $(m1,…, mi,…, ms) + (m1,…, ni,…, ns) = (m1+n1 ,…, mi+ni,…, ms+ns)$ $and r(m1,…, mi,…, ms) = (rm1,…, rmi,…, rms) ,$ where ${m}_{i},{n}_{i}\in {M}_{i}$, $r\in R$, and ${m}_{i}+{n}_{i}$ and $r{m}_{i}$ are given by the operations in ${M}_{i}$.

HW: Show that these are good definitions, i.e. that, as defined above, $M\oplus N$ and ${M}_{1}\oplus {M}_{2}\oplus \cdots \oplus {M}_{n}$ are left $R$-modules with zeros given by $\left({0}_{M},{0}_{N}$ and $\left({0}_{{M}_{1}},\dots ,{0}_{{M}_{s}}$, respectively. (${0}_{{M}_{i}}$ denotes the zero element in the left $R$-module ${M}_{i}$.)

## Notes and References

These notes are written to highlight the analogy between groups and group actions, rings and modules, and fields and vector spaces.

## References

[Ram] A. Ram, Notes in abstract algebra, University of Wisconsin, Madison 1993-1994.

[Bou] N. Bourbaki, Algèbre, Chapitre 9: Formes sesquilinéaires et formes quadratiques, Actualités Sci. Ind. no. 1272 Hermann, Paris, 1959, 211 pp. MR0107661.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.