2.2 Modules

Definition

Let $R$ be a commutative ring. An $R$-module $M$ is an additive abelian group with a function $R\times M\to M$, denoted as $(r,a)\to ra$ such that $\forall r,s\in R$, $a,b\in M$, we have

• $r(a+b)=ra+rb$;
• $(r+s)a=ra+sa$;
• $(rs)a=r(sa)$;
• $1_{R}a=a$.

Examples.

• If $R$ is a field, then an $R$-module is called a vector space.
• $\mathbb{Z}$-module iff abelian group.
• $f:R\to S$ is a homomorphism of rings, then any $S$-module can be regarded as an $R$-module, using $(r,m)=f(r)\cdot m$.
• An ideal $I\subseteq R$ is an $R$-module and $R/I$ is an $R$-module.

Remark. $0_R\cdot m=0_M$ for any $m\in M$ and $r\cdot 0_M=0_M$.

Definition

Let $M,N$ be two $R$-modules. Then $f:M\to N$ is called a homomorphism of $R$-module if it is a homomorphism of abelian groups and $f(rm)=rf(m)$. In this case, $\ker f\subseteq M$ and $\mathrm{Im}f\subseteq N$ are submodules.

Theorem

Let $N\subseteq M$ be an $M$-submodule. Then $M/N$ is also an $R$-module, using $(r,m+N)\mapsto rm+N$.

Theorem

Let $A$ be an $R$-module, and let $B,C\subseteq A$ be $R$-submodules.

• $B/(B\cap C)\simeq (B+C)/C$;
• if $C\subseteq B\subseteq A$, then $B/C\subseteq A/C$ is an $R$-submodule and $A/C/B/C\simeq A/B$.

Definition

• Let $A\stackrel{f}{\to }B\stackrel{g}{\to }C$ be a chain and homomorphism of $R$-modules. We say it is exact at $B$ if $\mathrm{Im}f=\ker g$.
• A finite sequence $A_0\stackrel{f_1}{\to }{A}_1\stackrel{f_2}{\to }\cdots \stackrel{f_n}{\to }{A}_n$ is exact, if it is exact at each $A_i(1⩽i⩽n-1)$, that is, $\mathrm{Im}{f}_i=\ker f_{i+1}$ for any $i$.
• If $0\stackrel{0}{\to }X\stackrel{f}{\to }Y\stackrel{g}{\to }Z\stackrel{0}{\to }0$ is exact, then we say it is short exact.

Remark.

• iii) is a short exact sequence iff $f$ is injective and $g$ is surjective.
• $I\subseteq R$ is an ideal, then $0\to I\to R\to R/I\to 0$ is exact.
• Define $\mathrm{coker}f=N/\mathrm{Im}f$ and $\mathrm{coim}f=M/\ker f$. Then $0\to \mathrm{Im}f\to N\to \mathrm{coker}f\to 0$ and $0\to \ker f\to M\to \mathrm{coim}f\to 0$ are exact.

short five lemma

Suppose we have a commutative diagram

$$\begin{array}{ccccccccc}0& \to & A& \stackrel{f}{\to }& B& \stackrel{g}{\to }& C& \to & 0\\ & & \alpha \downarrow & & \beta \downarrow & & \gamma \downarrow & & \\ 0& \to & A^{\prime }& \stackrel{f^{\prime }}{\to }& B^{\prime }& \stackrel{g^{\prime }}{\to }& C^{\prime }& \to & 0\end{array}$$

then $\alpha ,\gamma$ are monomorphism (rep. epimorphism, isomorphism) yield that $\beta$ is monomorphism (rep. epimorphism, isomorphism).

\begin{proof} 高辉说这辈子证明一次就够了。我就不证了。 \end{proof}

Remark. Short five lemma is a special case of snake lemma.

Definition

• If $0\to A\stackrel{f}{\to }B\stackrel{g}{\to }C$ is exact, then set it is a left exact sequence;
• If $A\stackrel{f}{\to }B\stackrel{g}{\to }C\to 0$ is exact, then set it is a right exact sequence.

Theorem

Let $0\to A\stackrel{f}{\to }B\stackrel{g}{\to }C\to 0$ be a short exact sequence, then TFAE:

• there exists $h:C\to B$ such that $g\circ h={\mathrm{id}}_C$;
• there exists $k:B\to A$ such that $k\circ f={\mathrm{id}}_A$;
• there exists a commutative diagram and so $B\simeq A\oplus C$.

$$\begin{array}{ccccccccc}0& \to & A& \stackrel{f}{\to }& B& \stackrel{g}{\to }& C& \to & 0\\ & & =\downarrow & & s\downarrow & & =\downarrow & & \\ 0& \to & A& \stackrel{{\iota }_1}{\to }& A\oplus C& \stackrel{{\iota }_2}{\to }& C^{\prime }& \to & 0\end{array}$$

\begin{proof} iii)→i). It is easy, as $h:=s^{-1}\circ {\iota }_2\circ {\mathrm{id}}_C$.

i)→iii). Define $t:A\oplus C\to B,(a,c)\mapsto f(a)+h(c)$. It is easy to verify when $s:=t^{-1}$ the diagram is commutative. \end{proof}