07. Modules - Definitions and Examples

Module

Let $R$ be a ring with identity $1_R$

Module over $R$ is abelian group $(M,+)$ with

Multiplication Action $a:R\times M\to M$ of $R$ on $M$ written as

$$(r,m)\mapsto r.m$$

which satisfies

$$1_R.m=m\text{ for all }m\in M$$

$$(r_1.r_2).m=r_1.(r_2.m)\text{ for all }{r}_1,r_2\in R,m\in M$$

$$(r_1+r_2).m=r_1.m+r_2.m\text{ for all }{r}_1,r_2\in R\text{ and }m\in M$$

$$r.(m_1+m_2)=r.m_1+r.m_2\text{ for all }r\in R\text{ and }{m}_1,m_2\in M$$

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Modules as Ring Homomorphisms into Endomorphism Rings

Let $M$ be an abelian group and ring $R$

1. Set $\mathrm{End}(M)=\mathrm{Hom}(M,M)$ of group homomorphisms from $M$ to itself is naturally a ring
   where addition is given pointwise and multiplication is given by composition

2. Giving $M$ the structure of an $R$-module through action $R\times M\to M$ is equivalent to
   Giving ring homomorphism $\varphi :R\to \mathrm{End}(M)$

Proof

As $M$ is abelian
If $f_1,f_2\in \mathrm{End}(M)$ then $f_1+f_2$ is a group homomorphism
As $f_1+f_2=f_2+f_1$ then addition in $\mathrm{End}(M)$ is commutative
Composition of functions gives $\mathrm{End}(M)$ a multiplication which distributes over addition
Hence $\mathrm{End}(M)$ is a ring

If map $a:RxM$ denoted by $(r,m)\mapsto r.m$ then equation $\varphi (r)(m)=r.m$
defines $\varphi$ in terms of $\varphi$ and $a$ conversely
Action $m\mapsto r.m$ is a homomorphism of abelian group $M$
with compatibility of $\varphi$ with multiplication and addition

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7.1 Submodules, Generation and Linear Independence

Submodule

Let $M$ be an $R$-module

Let $N\subseteq M$ then

$N$ is a submodule if it is an abelian subgroup of $M$ and for all $r\in R$ and $n\in N$ then

$$r.n\in N$$

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Submodule Generated by Subset

Let $X$ be an subset of an $R$-module $M$ then

Submodule generated or spanned by $X$ is

$$⟨X{⟩}_R=⟨X⟩=\bigcap _{N\supseteq X}N$$

where $N$ goes over submodules of $M$ containing $X$

Equivalent Form

$$⟨X⟩=R.X=\left\{\sum _{i=1}^k{r}_i{x}_i:r_i\in R,x_i\in X\right\}$$

Sum of Submodules

Let $N_1,N_2$ be submodules then

$$N_1+N_2=\{m+n:m\in N_1,n\in N_2\}$$

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Linearly Independent of a Set of a Module

Let $M$ be a module over $R$
Let set $S\subseteq M$ then

$S$ is linearly independent if

$$r_1{s}_1+r_2{s}_2+\cdots +r_k{s}_k=0\text{ where }{r}_i\in R,s_i\in S\text{ for }1\le i\le k$$

then

$$r_1=r_2=\cdots =r_k=0$$

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Basis of Module

Let $M$ be a module over $R$
Let set $S\subseteq M$ then

$S$ is a basis for $M$ if and only if it is

1. Linearly Independent
2. Spans $M$

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Free Module

Let $M$ be a module over $R$

$M$ is a Free module if it has a basis

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Freely Generated Module

Let $M$ be a module over $R$

$M$ is finitely generated if it is generated by some finite subset

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