01 - Rings

Ring

Ring is a datum

$$(R+,\times ,0,1)$$

where $R$ is a set and $1,0\in \mathbb{R}$ with binary operations $+,\times$ on $R$ such that

1. $R$ is an Abelian Group under $+$ with identity $0$

2. Binary operation $\times$ is associative and $1\times x=x\times 1=x$ for all $x\in R$

3. Multiplication distributes over addition

$$\begin{array}{rl}x\times (y+z)& =(x\times y)+(x\times z)\\ (x+y)\times z& =(x\times z)+(y\times z)\forall x,y,z\in R\end{array}$$

For real numbers or integer, tend to use $\cdot$ instead of $\times$

Commutative Ring

Ring with property

$$x\cdot y=y\cdot x\forall x,y\in R$$

Examples of Rings

1. Integers of $\mathbb{Z}$
2. Modulo arithmetic $\mathbb{Z}/n\mathbb{Z}$ for some $n\in \mathbb{Z}$ (integers modulo $n$)
3. Gaussian Integrals

$$\mathbb{Z}[i]=\{a+ib:a,b\in \mathbb{Z}\}$$

Subring

Let $R$ be a ring then

Subset $S\subseteq R$ is a subring if

1. $S$ inherits the structure of the ring from $R$ with $0,1\in S$
2. $S$ is closed under the addition and multiplication operations in $R$
3. $(S,+,\times ,0,1)$ is satisfies the axioms for a ring

Subring Criterion lemma

Let $R$ be a ring and $S$ a subset of $R$ then

$S$ is a subring if and only if

1. $1\in S$
2. For all $s_1,s_2\in S$ then

$$s_1{s}_2,s_1-s_2\in S$$

Proof

$S$ is non-empty as $1\in S$
Condition $s_1-s_2\in S$ for all $s_1,s_2\in S$ is to show $S$ is an additive subgroup (subgroup test)
Other conditions for the subring hold directly

Ring Homomorphism

Map $f:R\to S$ between rings $R$ and $S$ is a ring homomorphism if

1. $f(1_R)=1_S$
2. $f(r_1+r_2)=f(r_1)+f(r_2)$
3. $f(r_1\cdot r_2)=f(r_1)\cdot f(r_2)$

Inclusion Map

Let $R$ be a ring with subring $S$ then

Define Inclusion Map by

$$i:S\to R\text{ by }i(s)=s$$

where $1$ is the identity polynomial

Characteristic

Let $R$ be a ring then
Let $n_R\equiv \underset{n\text{ times}}{\underbrace{1+1+\cdots +1}}$
Let $d$ be the characteristic of $R$ then

$d$ is the smallest integer such that

$$d_R=\underset{d\text{ times }}{\underbrace{1+1+\cdots +1}}=0$$

where $1$ is the multiplicative identity of ring $R$

Ring of Sequences

Let $R$ be a sequence then

$$R^{\mathbb{N}}=\{f:\mathbb{N}\to R\}$$

is the set of sequences

Convolution of Ring on $R^{\mathbb{N}}$

Let $R$ be a ring

Consider Discrete Convolution $(f,g)\mapsto f\ast g$
For $f,g\in S$ then function

$$(f\ast g)(n)=\sum _{k\in \mathbb{Z}}f(k)g(n-k)$$

is well-defined

Convolution of Ring

Let $R$ be a ring

Let $S=\{f:\mathbb{Z}\to R:\exists N\in \mathbb{Z}\text{ such that }f(n)=0\forall n<N\}$

For $f,g\in S$ then function

$$(f\ast g)(n)=\sum _{k\in \mathbb{Z}}f(k)g(n-k)$$

is well-defined

Evaluation Homomorphism lemma

Let $R$ and $S$ be rings
Homomorphism $\Phi :R[t]\to S$ is determined by pair

$$(\psi ,s)\text{ where }\varphi ={\Phi }_{\mid R}\text{ and }s=\Phi (t)$$

where any such pair determines unique ring homomorphism

Hence set of all ring homomorphisms

$$\{\Phi :R[t]\to S\}$$

is bijective with set of all pairs $\{(\varphi ,s):\Phi :R\to R,s\in S\}$

Proof

Element of $R[t]$ is in form

$$\sum _{i=0}^n{a}_i{t}^i\text{ where }{a}_i\in R$$

Hence if $\Theta$ is any homomorphism satisfying

$$\Theta \circ i=\varphi \text{ and }\Theta (t)=s$$

where $i$ is the Natural inclusion map

Then

$$\Theta \left(\sum _{i=0}^n{a}_i{t}^i\right)=\sum _{i=0}^n\Theta (a_i{t}^i)=\sum _{i=0}^n\Theta (a_i)\Theta (t^i)=\sum _{i=0}^n\varphi (a_1)s^i$$

Hence $\Theta$ is uniquely determined

For any pair $\varphi ,s$ there is corresponding homomorphism

$$\Phi :R[t]\to s$$

then require

$$\Phi \left(\sum _{i=0}^n{a}_i{t}^i\right)=\sum _{i=0}^n\varphi (a_i)s^i$$

to be a homomorphism which it is from definitions

Zero Divisor

Let $R$ be a commutative ring then

Element $a\in R\setminus \{0\}$ is a zero divisor if

$$\exists b\in R\setminus \{0\}\text{ such that }a.b=0$$

Units of Ring

Let $R$ be a ring then subset

$$R^{\times }=\{r\in R:\exists s\in R\text{ such that }r.s=1\}$$

Kernel of Ring

Let $f:R\to S$ be a ring homomorphism then

Kernel of $f$ is

$$\ker (f)=\{r\in R:f(r)=0\}$$

Image of Ring

Let $f:R\to S$ be a ring homomorphism then

Image of $f$ is

$$\mathrm{im}(f)=\{s\in S:\exists r\in R\text{ such that }f(r)=s\}$$

Associates

Let $a,b\in R$ where $R$ is a ring then

$a,b$ are associates if

$$\exists u\in {\mathbb{R}}^{\times }\text{ such that }a=u.b$$

Note that it is an equivalence class

Degree of Polynomials of Ring

Let $R$ be a ring

Suppose $f\in R[t]$ is non-zero then

$$f=\sum _{i=0}^n{a}_i{t}^i$$

where $a_n\ne 0$ then

$$\mathrm{deg}(f)=n$$

where $a_n$ is the leading coefficient of $f$

Degree of Product of Polynomials in Integral Domain

Let $R$ be an integral domain then

For $f,g\in R[t]$ then

$$\mathrm{deg}(f.g)=\mathrm{deg}(f)+\mathrm{deg}(g)$$

Note that this implies $R[t]$ is also an integral domain

Division Algorithm lemma

Let $R$ be a ring with

$$f=\sum _{i=0}^n{a}_i{t}^i\in R[t]\text{ where }{a}_n\in R^{\times }$$

Then if $g\in R[t]$ is any polynomial then

Exists $q,r\in R[t]$ such that

$$r=0\text{ or }\mathrm{deg}(r)<\mathrm{deg}(f)\text{ and }g=q.f+r$$

Proof

Induction on $\mathrm{deg}g$
As $a_n\in R^{\times }$

If $h\in R[t]\setminus \{0\}$ then

$$\mathrm{deg}(f.h)=\mathrm{deg}(f)+\mathrm{deg}(h)$$

Hence if $\mathrm{deg}(g)<\mathrm{deg}(f)$ then $q=0$ and hence $r=g$

Otherwise suppose

$$g=\sum _{j=0}^m{b}_j{t}^j\text{ where }{b}_m\ne 0\text{ and }m=\mathrm{deg}(g)\ge n\ge \mathrm{deg}(f)$$

Then $a_n^{-1}{b}_m{t}^{m-n}.f$ has leading erm $b_m{t}^m$

has leading term $b_m{t}^m$ has leading term $b_n{t}^m$
As

$$h=g-a_n^{-1}{b}_m{t}^{m-n}f$$

as $\mathrm{deg}(h)<\mathrm{deg}(q)$

Then by induction there exists unique $q^{\prime },r^{\prime }$ such that $h=q^{\prime }.f+r^{\prime }$
Setting

$$q=a_n^{-1}{b}_n{t}^{m-n}+q^{\prime }\text{ and }r=r^{\prime }$$

then

$$g=q.f+r$$

As $q$ and $r$ are uniquely determined by $q^{\prime }$ and $r^{\prime }$ then they are also unique