• Abstract Algebra
• integer
  • natural number
  • well-ordering axiom
  • division algorithm
    • divisibility
    • greatest common divisor
      • Euclidean algorithm
      • Bézout’s identity
      • relatively prime
    • prime number
      • Euclid’s lemma
      • fundamental theorem of arithmetic
• relation
  • partition of set
  • equivalence relation
    • equivalence class
  • modular arithmetic
    • congruence module $n$
    • congruence class
    • addition and multiplication in modular arithmetic
      • unit
      • zero divisor
• group
  • abelian group (commutative group)
  • nonabelian group
    • general linear group of degree $n$ over $R$
      • special linear group of degree $n$ over $R$
  • binary operation
    • associative binary operation
    • commutative binary operation
  • order of group
    • order of element in group
  • Torsion group
  • dihedral group $D_{2n}$
    • permutation of set
  • symmetric group on set
    • array notation of permutation
    • cycle notation of permutation
      • transposition
      • disjoint cycle
  • subgroup
    • subgroup test
    • centralizer of group
      • center of group
    • normalizer of group
  • cyclic group
    • fundamental theorem of cyclic group
  • homomorphism
    • well-defined map
    • automorphism
    • isomorphism
      • equivalence relation from isomorphism
      • cyclic isomorphism
      • Cayley’s theorem
    • kernel and image of homomorphism
  • coset
    • set of coset
      • index of subgroup in group
      • Lagrange’s theorem
  • normal subgroup
    • normal subgroup test
  • quotient subgroup
    • natural projection
  • isomorphism theorem
    • first isomorphism theorem
    • second isomorphism theorem (diamond theorem)
    • third isomorphism theorem
    • forth isomorphism theorem
• ring
  • commutative ring
  • ring with identity
  • division ring
  • field
  • zero divisor of ring
    • integral domain
  • unit of ring
  • subring
    • subring test
    • center of ring
    • characteristic of ring
    • nilpotent of commutative ring with identity
  • polynomial ring
    • polynomial in $x$ with coefficient in $R$
  • matrix ring
    • scalar matrix
  • ring homomorphism
  • ideal of ring
    • ideal test
    • maximal ideal
    • prime ideal
  • quotient ring
  • isomorphism theorem for ring
    • first isomorphism theorem for ring
    • second isomorphism theorem for ring
    • third isomorphism theorem for ring
    • forth isomorphism theorem for ring

Abstract Algebra

integer

$$\mathbb{Z}:=\{\cdots ,-2,-1,0,-1,-2,\cdots \}$$

natural number

$$\mathbb{N}:=\{0,1,2,\cdots \}$$

well-ordering axiom

$\forall R\subseteq \mathbb{N},R\ne \varnothing ,\exists a\in R$ s.t. $\min (R)=a$

division algorithm

$\forall a,b\in \mathbb{Z},b>0,\exists !q,r\in \mathbb{Z}$, s.t.

$$a=bq+r,0\le r<b$$

divisibility

$b|a$, or $b$ divide $a\iff \exists c\in \mathbb{Z}$, s.t. $a=bc$

• $b\nmid a$, or $b$ do not divide $a$
• $b|a\iff b|(-a)$
• $a\ne 0,b|a\Rightarrow b\le |a|$

greatest common divisor

$a,b\in \mathbb{Z}\setminus \{0\},\exists !d:=\gcd (a,b)$, s.t.

1. $d|a,d|b$
2. $c|a,c|b\Rightarrow c\le d$

or

1. $d|a,d|b$
2. $c|a,c|b\Rightarrow c|d$

Euclidean algorithm

$a,b\in \mathbb{Z}\setminus \{0\}$

1. let $r_{-2}=a,r_{-1}=b$

2. apply division algorithm on $r_{i-1},r_i$ until $r_{n+1}=0$

   $$r_{i-1}=q_{i+1}{r}_i+r_{i+1}$$

3. $r_n=\gcd (a,b)$

• $a,b,q,r\in \mathbb{Z},b>0,a=bq+r\Rightarrow \gcd (a,b)=\gcd (b,r)$

Bézout’s identity

$a,b\in \mathbb{Z}\setminus \{0\},\exists u,v$ s.t.

$$\gcd (a,b)=au+bv$$

• $\gcd (a,b)=\min (\{ax+by|x,y\in \mathbb{Z}\})$
• $x,y\in \mathbb{Z},c=ax+by\Rightarrow \gcd (a,b)|c$
• find $u,v$: substitute $\gcd (a,b)$ from the last equation in Euclidean algorithm up

relatively prime

$a,b$ are relatively prime $\iff \gcd (a,b)=1$

• $\gcd (a,b)=1\iff \exists x,y\in \mathbb{Z},ax+by=1$
• $a|bc,\gcd (a,b)=1\Rightarrow a|c$

prime number

$p\in \mathbb{Z}$, then $p$ is prime $\iff p\ne 0,p\ne \pm 1$, the only divisor of $p$ are $\pm 1,\pm p$

• composite number: not prime

Euclid’s lemma

$p$ prime, $p|ab\Rightarrow p|a$ or $p|b$

• $p\in \mathbb{Z}\setminus \{0,\pm 1\}$, then $p$ prime $\iff$ if $p|ab$, then $p|a$ or $p|b$
• $p$ prime, $p|a_1{a}_2\cdots a_n\Rightarrow \exists i,p|a_i$

fundamental theorem of arithmetic

$\forall n\in \mathbb{Z}\setminus \{0,\pm 1\},n$ can be factored uniquely into product of primes

relation

relation $R$ on $A$ is a subset $R\subseteq A\times A$

$aRb$ or $a$ is related to $b\iff (a,b)\in R$

• $a/Rb$ or $a$ is not related to $b$

partition of set

$A$ is a non-empty set, partition of $A$

$$\mathcal{P}:\{A_i|i=1,2,\cdots \}$$

s.t.

1. $\forall i\ne j,A_i\cap A_j=\varnothing$
2. $A=\bigcup _i{A}_i$

• $R_{\mathcal{P}}:=\{(a,b)|\exists A_i,a\in A_i,b\in A_i\}$ is equivalence relation

equivalence relation

1. reflexive: $aRa$
2. symmetric: $aRb\Rightarrow bRa$
3. transitive: $aRb,bRc\Rightarrow aRc$

equivalence class

equivalence class of representative $a$

$$[a]:=\{x\in A|xRa\}$$

• $aRb\iff [a]=[b]$

• $[a],[b]$ are either disjoint or identical

• quotient set of $A$ by $R$

  $$A/R:=\{[a]|a\in A\}$$

  is partition of $R$

modular arithmetic

$a,b,n\in \mathbb{Z},n>0$, then

$a{\equiv }_{n}b$ or $a\equiv b$ (mod $n$) or $a$ is congruent to $b$ modulo $n\iff n|(a-b)$

congruence module $n$

$\mathbb{Z}/n$ ($\mathbb{Z}$ mod $n$) or quotient set of $\mathbb{Z}$ by ${\equiv }_n$

• $\mathbb{Z}/n$ is equivalence relation on $\mathbb{Z}$

congruence class

equivalence class of $a$ modulo $n$

$$\begin{gathered} [a{]}_n:=\{x\in \mathbb{Z}|x{\equiv }_{n}a\} \\ =\{a+nk|k\in \mathbb{Z}\} \end{gathered}$$

• there are $n$ distinct congruence classes

addition and multiplication in modular arithmetic

$$\begin{gathered} [a]+[b]=[a+b] \\ [a][b]=[ab] \end{gathered}$$

• addition and multiplication are well-defined
• $\forall [a]\in \mathbb{Z}/n,\exists !$ additive inverse
• $[a]\ne [0]$ either has unique multiplicative inverse or it has not

unit

$\exists x\in \mathbb{Z}/n,[a]x=[1]$

• $[a]$ is unit $\iff \gcd (a,n)=1$

zero divisor

$\exists x\ne [0],[a]x=[0]$

• $p\in \mathbb{N}\setminus \{0,1\}$, then

  $p$ is prime

  $\iff \forall [a]\ne [0]$, $[a]$ is unit

  $\iff [b][c]=[0]\Rightarrow [b]=[0]$ or $[c]=[0]$, or no zero divisor

group

ordered pair $(G,\ast )$

• set $G$
• binary operation $\ast$

axiom group satisfy:

1. $\ast$ is associative
2. identity: $\exists e\in G,\forall a\in G,a\ast e=e\ast a=a$
3. inverse: $\forall a\in G,\exists a^{-1}\in G,a\ast a^{-1}=a^{-1}\ast a=e$

property

1. $e$ is unique
2. $\forall a\in G$, $a$ has unique inverse
3. $a\ast x=b$ and $y\ast a=b$ has unique solution, cancellation law work

• $(a^{-1}{)}^{-1}=a$
• $(a\ast b{)}^{-1}=b^{-1}\ast a^{-1}$
• value of $a_1\ast a_2\ast \cdots \ast a_n$ is independent of the bracketing

abelian group (commutative group)

$\ast$ is commutative

• $\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C},\mathbb{Z}/n,{\mathbb{R}}^{m\times n}$ under $+$
• $\mathbb{Q}\setminus \{0\},\mathbb{R}\setminus \{0\},\mathbb{C}\setminus \{0\},(\mathbb{Z}/n{)}^{\times }$ under $\cdot$

nonabelian group

• dihedral group of order $2n$, $D_{2n}$
• $(GL(n,\mathbb{R}),\cdot )$

general linear group of degree $n$ over $R$

$$GL(n,R):=\{A\in R^{n\times n}|A\text{ is invertible}\}$$

special linear group of degree $n$ over $R$

$$SL(n,R):=\{A\in GL(n,R)|\det A=1\}$$

binary operation

$$\ast :G\times G\to G$$

associative binary operation

$$(a\ast b)\ast c=a\ast (b\ast c)$$

commutative binary operation

$$a\ast b=b\ast a$$

order of group

$|G|$, or order of $G$, is the number of distinct elements in $G$

order of element in group

$|a|$, or order of $a$, is the smallest $n\in {\mathbb{Z}}_{+}$ s.t.

$$a^n=e$$

or $\infty$ if $n$ DNE

• $e,a,a^2,\cdots ,a^{n-1}$ are distinct
• $a^m=1\iff n|m$
• $\forall m\in {\mathbb{Z}}_{+},|a^m|=\frac{n}{\gcd (m,n)}$

Torsion group

$G$ is a Torsion group, then $\forall a\in G,|a|$ is finite

• Torsion-free group: $\forall a\in G,a\ne e,|a|=\infty$

• Torsion subgroup of $G$: for abelian group $G$,

  $$\text{Tor}(G):=\{a\in G||a|\text{ is finite}\}$$

• $ab=ba,\gcd (|a|,|b|)=1\Rightarrow |ab|=|a||b|$

• $G$ is abelian Torsion group, then $c$ has largest order $\Rightarrow \forall a\in G,|a|||c|$

dihedral group $D_{2n}$

rotation ($r$) and reflection ($s$) of n-gon

$$\begin{gathered} \{1,s,r,sr,r^2,s{r}^2,\cdots ,r^{n-1},s{r}^{n-1}\}n\ge 3 \\ (r^i)(k)=i+k\mathrm{mod}n \end{gathered}$$

• order $2n$
• degree $n$
• $|s|=2$
• $|r|=n$
• $\forall 0\le i\ne j\le n-1,s{r}^i\ne s{r}^j$
• $\forall 0\le i\le n,r^{i}s=s{r}^{-i}$

permutation of set

permutation of $X$: bijection

$$\sigma :X\to X$$

symmetric group on set

set $\Omega \ne \varnothing$

$S_{\Omega }$, the set of permutation of $\Omega$

$(S_{\Omega },\circ )$, symmetric group on $\Omega$

• $n\ge 1,\Omega =X_n\Rightarrow (S_n,\circ )$, symmetric group of degree $n$
• $|S_{\Omega }|=n!$
• $n\ge 3\Rightarrow (S_n,\circ )$ nonabelian

array notation of permutation

$\alpha$ is permutation on $X_n$

$$\alpha =\left(\begin{array}{cccc}1& 2& \cdots & n\\ \alpha (1)& \alpha (2)& \cdots & \alpha (n)\end{array}\right)$$

cycle notation of permutation

cyclically permute and fix the rest

$$(a_1{a}_2\cdots a_m)=\left(\begin{array}{cccc}{a}_1& a_2& \cdots & a_m\\ a_2& a_3& \cdots & a_1\end{array}\right)$$

• $m$-cycle (or cycle of length $m$)

transposition

$2$-cycle

• $\forall \sigma \in S_n,\sigma$ can be expressed as product of transposition
  • not unique
  • even permutation: length of such transposition is even
  • odd permutation

disjoint cycle

no common number

• $\alpha ,\beta \in S_n$ disjoint $\Rightarrow \alpha \beta =\beta \alpha$
• $\forall \pi \in S_n,\pi \ne 1,\pi$ can be uniquely expressed as product of disjoint cycle of length at least 2

subgroup

$H\le G$, or $H\subseteq G$ is a subgroup of $G$

1. $H\ne \varnothing$
2. $\forall x,y\in H,xy\in H$
3. $\forall x\in H,x^{-1}\in H$

• $H<G$, or $H$ is a proper subgroup of $G$

subgroup test

1. $H\le G\iff$
   1. $H\ne \varnothing$
   2. $\forall x,y\in H,x{y}^{-1}\in H$
2. $|G|<\infty ,H\le G\iff$
   1. $H\ne \varnothing$
   2. $\forall x,y\in H,xy\in H$

centralizer of group

$$C_G(A):=\{g\in G|\forall a\in A,ga=ag\}$$

• $C_G(a)$ if $A=\{a\}$

center of group

$$Z(G):=\{g\in G|\forall a\in G,ga=ag\}$$

normalizer of group

$$N_G(A):=\{g\in G|gA{g}^{-1}=A\}$$

where

$$gA{g}^{-1}:=\{ga{g}^{-1}|a\in A\}$$

for group $G$ and subset $A\subseteq G$

• $C_G(A)\le N_G(A)\le G$
• $G$ abelian $\Rightarrow Z(G)=G,C_G(A)=N_G(A)$

cyclic group

$$G=⟨x⟩=\{x^n|n\in \mathbb{Z}\}$$

• generator $x$
• cyclic group are abelian
• $|G|=|x|$

fundamental theorem of cyclic group

1. $H\le G\Rightarrow \exists !k\in \mathbb{N},k={\min }_k\{x^k\in H\},H=⟨x^k⟩$
2. $|G|=n\Rightarrow$
   • $\forall a\in \mathbb{N},a|n,\exists !A,|A|=a$
     • $A=⟨x^d⟩$
     • $d=\frac{n}{a}$
   • $\forall m\in \mathbb{Z},⟨x^m⟩=⟨x^{\gcd (m,n)}⟩$
   • $\forall i,j,x^i=x^j\iff i\equiv j\mathrm{mod}n$
3. $|G|=\infty \Rightarrow \forall m\in \mathbb{Z},⟨x^m⟩=⟨x^{|m|}⟩$

homomorphism

$(G,\cdot ),(H,\cdot )$ are group

well-defined map $\phi :G\to H$ is homomorphism $\iff$

$$\forall a,b\in G,\phi (ab)=\phi (a)\phi (b)$$

property

1. $\phi (1_G)=1_H$
2. $\forall n\in \mathbb{Z},\phi (a^n)=\phi (a{)}^n$

• monomorphism: $\phi$ injective
• epimorphism: $\phi$ surjective
• $\cong$, or isomorphism: $\phi$ bijective

well-defined map

$f:A\to B$ is well-defined $\iff$

$$\forall x=y\in A\Rightarrow f(x)=f(y)$$

automorphism

isomorphism from $G$ to $G$

• inner automorphism of $G$, ${\phi }_a(g)=ag{a}^{-1}(a\in G)$

isomorphism

property

1. $|G|=|H|$
2. $G$ abelian $\iff H$ abelian
3. $\forall g\in G,|g|=|\phi (g)|$
4. $\forall n\in {\mathbb{Z}}^{+},G,H$ have same number of elements of order $n$

• $|G|=p$ prime $\Rightarrow G\cong \mathbb{Z}/p$

equivalence relation from isomorphism

$\mathcal{G}$ is set of groups

$\cong$ is equivalence relation on $\mathcal{G}$

cyclic isomorphism

$G=⟨x⟩$

• $|x|=\infty \Rightarrow G\cong \mathbb{Z}$
• $|x|=n\Rightarrow G\cong \mathbb{Z}/n$

Cayley’s theorem

$\forall$ group $G$, $\exists$ permutation $H\le S_G$, s.t. $G\cong H$

kernel and image of homomorphism

$\phi :G\to H$ is homomorphism

$h\in H$

1. kernel of $\phi$

   $$\text{Ker }\phi :=\{a\in G|\phi (a)=1\}$$

   • $\text{Ker }\phi \le G$
   • measure how much $\phi$ is not injective
   • $\phi$ is injective $\iff \text{Ker }\phi =\{1\}$

2. image of $\phi$

   $$\text{Im }\phi =\phi (G):=\{\phi (a)|a\in G\}$$

   • $\text{Im }\phi \le H$
   • $\phi$ is injective $\iff \text{Im }\phi \cong G$

3. fiber of $h$ under $\phi$

   $${\phi }^{-1}(h):=\{a\in G|\phi (a)=h\}$$

   • $\phi$ is surjective $\iff \text{Im }\phi =H$

coset

$H\le (G,\cdot )$, $g,a,b\in G$

1. left coset of $H$ in $G$

   $$gH:=\{gh|h\in H\}$$

   • $\exists$ bijection between $H$ and $gH$
   • $aH=bH\iff b^{-1}a\in H$
   • $aH=bH$ or $aH\cap bH=\varnothing$

2. right coset of $H$ in $G$

   $$Hg:=\{hg|h\in H\}$$

   • $\exists$ bijection between $H$ and $Hg$
   • $Ha=Hb\iff a{b}^{-1}\in H$
   • $Ha=Hb$ or $Ha\cap Hb=\varnothing$

3. $g$ is representative

• left and right coset are not necessarily equal
• $gH\le H\iff g=1$
• $|H|=|gH|=|Hg|$

set of coset

1. set of left coset

   $$G/H:=\{aH|a\in G\}$$

2. set of right coset

   $$H\setminus G:=\{Ha|a\in G\}$$

• $\exists$ bijection between $G/H$ and $H\setminus G$

index of subgroup in group

index of $H$ in $G$

$$[G:H]$$

is number of distinct left (right) coset

Lagrange’s theorem

1. $|H|||G|$
2. $[G:H]=\frac{|G|}{|H|}$

for finite $G$

• $a\in G\Rightarrow |a|||G|$
• $\forall a\in G,a^{|G|}=1$

normal subgroup

$N⊴G$, or $N$ is normal in $G\iff$

$$\forall a\in G,aN=Na$$

normal subgroup test

1. $N⊴G$
2. $\iff \forall a\in G,aN{a}^{-1}\subseteq N$
3. $\iff \forall a\in G,aN{a}^{-1}=N$
4. $\iff N_G(N)=G$
5. $\iff$ if $aN=bN,cN=dN$, then $(ac)N=(bd)N$

quotient subgroup

$N⊴G$

$G/N=N\setminus G$ is the quotient group of $G$ by $N$

under operation $\cdot$:

$$(aN)(bN):=(ab)N$$

• $|G/N|=[G:N]$

natural projection

$N⊴G$

natural projection of $G$ onto $N$

homomorphism $\pi :G\to G/N$

$$\pi (a)=aN$$

• $\pi$ is epimorphism
• $\text{Ker }\pi =N$

isomorphism theorem

first isomorphism theorem

$\phi :G\to H$ homomorphism $\Rightarrow$

$$G/\text{Ker }\phi \cong \text{Im }\phi$$

• under $\Phi :G/\text{Ker }\phi \to \text{Im }\phi$

  $$\Phi (g\text{Ker }\phi ):=\phi (g)$$

second isomorphism theorem (diamond theorem)

$K\le G,N\le G,N⊴G\Rightarrow$

1. $KN\le G$
2. $K\cap N⊴K$
3. $N⊴KN$
4. $K/K\cap N\cong KN/N$
5. $[KN:K]=[N:K\cap N]$

third isomorphism theorem

$H⊴G,K⊴G,K\le H$

1. $H/K⊴G/K$
2. $(G/K)/(H/K)\cong G/H$

   • $G/K\cong G/H$ under $\Gamma :G/K\to G/H$

     $$\Gamma (gK):=gH$$

forth isomorphism theorem

$N⊴G$, natural projection $\pi :G\to G/N$

$\exists$ bijection $\Pi$

$$\begin{gathered} \Pi :\{A\le G|N\subseteq A\}\to \{\mathcal{A}\le G/N\} \\ \Pi (A)=\pi (A)=\{\pi (a)|a\in A\}=A/N \end{gathered}$$

$A,B\le G,N\le A,B$

property of $\Pi$

1. $A\le B\iff \Pi (A)\le \Pi (B)$
2. $A\le B\Rightarrow [B:A]=[\Pi (B):\Pi (A)]$
3. $\Pi (A\cap B)=\Pi (A)\cap \Pi (B)$
4. $A⊴G\iff \Pi (A)⊴G/N$

ring

a triple $(R,+,\cdot )$ s.t.

1. $(R,+)$ is abelian group

2. $\cdot$ is associative

   $$\forall a,b,c\in R,(ab)c=a(bc)$$

3. distributive

   $$\forall a,b,c\in R,a(b+c)=ab+ac$$

• $\forall a,b\in R,0a=a0=0$
• $\forall a,b\in R,(-a)b=a(-b)=-(ab)$
• $\forall a,b\in R,(-a)(-b)=ab$

commutative ring

$\iff \forall a,b\in R,ab=ba$

ring with identity

$\iff \exists 1\in R,\forall a\in R,a1=1a=a$

• $1$ is unique
• $\forall a\in R,-a=(-1)a$

division ring

$\iff (R,+,\cdot )$ is ring with identity and $\forall a\in R\setminus \{0\},\exists b,ab=ba=1$

field

commutative division ring

zero divisor of ring

$a\in R,a\ne 0$

$a$ is zero divisor $\iff$

$$\exists b\in R,b\ne 0\text{ s.t. }ab=0\text{ or }ba=0$$

• $\forall a,b,c\in R,a$ is not zero divisor, $ab=ac\Rightarrow a=0$ or $b=c$

integral domain

commutative ring $(R,+,\cdot ),1\ne 0$

$(R,+,\cdot )$ is integral domain $\iff$ no zero divisor:

$$\begin{gathered} \forall a,b,c\in R,ab=0\Rightarrow a=0\text{ or }b=0 \\ a\ne 0,b\ne 0\Rightarrow ab\ne 0 \end{gathered}$$

• $\forall a,b,c\in R,ab=ac\Rightarrow a=0$ or $b=c$
• $R$ is finite $\Rightarrow R$ is field

unit of ring

$R$ is ring with identity, $1\ne 0,a\in R$

$a$ is unit $\iff$

$$\exists b\in R\text{ s.t. }ab=1=ba$$

• group of units of $R$, $R^{\times }:=\{\text{units in }R\}$
• zero divisor cannot be unit

subring

$S\subseteq R$

$S$ is subring of $R\iff$

1. $(S,+)$ is a subgroup of $S$
2. $S$ is closed under multiplication

relationship between identity

• $1_S\not\equiv 1_R$
• $1_R\in S\Rightarrow 1_S=1_R$, unit in $S$ are unit in $R$

subring test

$S$ is subring of $R\iff$

1. $S\ne \varnothing$
2. $\forall a,b\in S,a-b\in S$
3. $\forall a,b\in S,ab\in S$

$S$ is finite subring of $R\iff$

1. $S\ne \varnothing$
2. $\forall a,b\in S,a+b\in S$
3. $\forall a,b\in S,ab\in S$

center of ring

$R$ is ring with $1$

$$Z(R):=\{z\in R|\forall r\in R,zr=rz\}$$

• is subring with $1$
• $R$ is division ring $\Rightarrow Z(R)$ is field

characteristic of ring

if $\exists n$ s.t.

$$n:=\underset{m}{\mathrm{argmin}}\{m\in {\mathbb{Z}}^{+}|\forall a\in R,ma=0\}$$

then $char(R)=n$, else $char(R)=0$

• $char(R)>0\iff char(R)=\underset{n}{\mathrm{argmin}}\{n\in {\mathbb{Z}}^{+}|n1=0\}$
• $R$ is integral domain $\Rightarrow char(R)$ is $0$ or prime
• $R$ finite $\Rightarrow char(R)||R|$

nilpotent of commutative ring with identity

commutative ring $R$ with $1$

$x$ is nilpotent $\iff$

$$\exists m\in {\mathbb{Z}}^{+}\text{ s.t. }{x}^m=0$$

• $x$ is $0$ or zero divisor
• $\forall r\in R,rx$ is nilpotent
• $1+x$ is unit
• $\forall$ unit $u,u+x$ is unit

polynomial ring

$R$ is commutative ring with identity, $x$ is indeterminate

ring of polynomial in $x$ with coefficient in $R$

$$R[x]:=\{\text{polynomial in x with coefficient in R}\}$$

• commutative
• identity $1$
• $R$ is subring of $R[x]$
• $S$ is subring of $R\Rightarrow S[x]$ is subring of $R[x]$
• $R$ is integral domain, $p(x),q(x)\in R[x]\setminus \{0\}\Rightarrow$
  1. $R[x]$ is integral domain
  2. $\mathrm{deg}(p(x)q(x))=\mathrm{deg}(p(x))\mathrm{deg}(q(x))$
  3. $(R[x]{)}^{\times }=R^{\times }$

polynomial in $x$ with coefficient in $R$

$n\in \mathbb{N},a_i\in R,a_n\ne 0$

$$p(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$$

• degree $\mathrm{deg}(p(x))=n$

matrix ring

$R$ is ring, $n\in {\mathbb{Z}}^{+}$

matrix ring of degree $n$ over $R$

$$M_n(R):=R^{n\times n}$$

• $R$ has identity $\Rightarrow 1_{M_n(R)}={\mathbb{I}}_n$

$n\ge 2\Rightarrow$

• $M_n(R)$ is not commutative
• $M_n(R)$ has zero divisor
• set of scalar matrix in $M_n(R)$ is a subring
• $S$ is subring of $R\Rightarrow M_n(S)$ is subring of $M_n(R)$

scalar matrix

$\forall i\ne j,a_{ii}=a,a_{ij}=0$

ring homomorphism

$R,S$ are ring

well-defined map $\phi :R\to S$ is ring homomorphism $\iff \forall r,s\in R,$

1. $\phi (r+s)=\phi (r)+\phi (s)$
2. $\phi (rs)=\phi (r)\phi (s)$

• monomorphism, epimorphism, isomorphism, kernel, image
• $\text{Ker}\phi$ is subring
• $\text{Im}\phi$ is subring
• $R\cong S\Rightarrow$ they have the same property (commutative, identity, inverse)

ideal of ring

subring $I$ is ideal of $R\iff$

$$\forall r\in R,a\in I,ra\in I,ar\in I$$

• proper ideal: proper subset that is ideal

ideal test

subset $I$ of ring $R$

$I$ is ideal $\iff$

1. $I\ne \varnothing$
2. $\forall a,b\in I,a-b\in I$
3. $\forall r\in R,a\in I,ra\in I,ar\in I$

$\iff$

$$a+I=b+I,c+I=d+I\Rightarrow (ac)+I=(bd)+I$$

maximal ideal

ideal $M$ of $R$ is maximal ideal $\iff$

1. $M⊊R$
2. $I$ is ideal of $R$, $M\subseteq I\subseteq R\Rightarrow I=M$ or $I=R$

• easily seen in lattice
• not necessarily exist
• $1\in R\Rightarrow$ every proper ideal is contained in a maximal ideal
• $R$ is commutative, $1\in R\iff R/M$ is field

prime ideal

ideal $P$ of $R$ is prime ideal $\iff$

1. $P⊊R$
2. $\forall a,b\in R,ab\in P\Rightarrow a\in P$ or $b\in P$

• $R$ is commutative, $1\in R\Rightarrow$
  • $P$ is prime ideal $\iff R/P$ is integral domain
  • $R$ is integral domain $\iff \{0\}$ is prime ideal
  • every maximal ideal is prime ideal

quotient ring

$I$ is ideal of $R$

$R/I=R\setminus I$ is quotient ring of $R$ by $I$

• $1\in R\Rightarrow 1_{R/I}=1+I$

• $R$ commutative $\Rightarrow R/I$ commutative

• natural projection

  $$\pi (r):=r+I:R\to R/I$$

isomorphism theorem for ring

first isomorphism theorem for ring

$\phi :R\to S$ is ring homomorphism $\Rightarrow$

1. $\text{Ker}\phi$ is ideal of $R$
2. $R/\text{Ker}\phi \cong \text{Im }\phi$

second isomorphism theorem for ring

$A,B$ are subring of $R$, $B$ is ideal $\Rightarrow$

1. $A+B$ is subring of $R$
2. $B$ is ideal of $A+B$
3. $A\cap B$ is ideal of $A$
4. $A/A\cap B\cong (A+B)/B$

third isomorphism theorem for ring

$I,J$ are ideal of $R$, $I\subseteq J\Rightarrow$

1. $J/I$ is ideal of $R/I$
2. $(R/I)/(J/I)\cong R/J$

forth isomorphism theorem for ring

$I$ is ideal of $R$, natural projection $\pi :R\to R/I$

$\exists$ bijection

$$\begin{gathered} \Pi :\{S\text{ is subring of }R|S\supseteq I\}\to \{\mathcal{S}\text{ is subring of }R/I\} \\ \Pi (S)=\pi (S)=S/I \end{gathered}$$

$A,B$ are subring of $R$, $I\subseteq A,B\Rightarrow$

1. $A\subseteq B\iff A/I\subseteq B/I$
2. $A\subseteq B\iff [B:A]=[B/I:A/I]$
3. $(A\cap B)/I=(A/I)\cap (B/I)$
4. $A$ is ideal of $R\iff A/I$ is ideal of $R/I$