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content/notes/math/algebra.mdx

@@ -4,7 +4,7 @@ subject: 'Mathematics'
 showToc: true
 ---
 
-# Category
+# Category theory
 
 <LatexFig width={50} src='/fig/category.svg' alt=''
   caption='Category diagram'
@@ -259,24 +259,13 @@ Note that the closure property follows implicitly from the definition $*: G \tim
 A group $(G, *)$ is abelian if $a * b = b * a$ for all $a,b \in G$ (commutativity). 
 </MathBox>
 
-<MathBox title='Ring' boxType='definition'>
-A ring is a non-empty set $R$ together with two binary (closed) operations, $+: R \times R \to R$ (addition) and $\cdot: R\times R\to R$, with the properties
-- $(R, +)$ is an abelian group
-- $(R, \cdot)$ is a monoid
-- Multiplication is distributive related to addition, i.e. for all $a,b,c\in R$
-  - $(a + b)c = ac + bc$
-  - $c(a + b) = ca + cb$
-
-A ring is commutative if $ab = ba$ for all $a,b\in R$. If a ring $R$ contains an element $e$ such that $ae = ea = a$ for all $a\in R$, we say that $R$ is a ring with identity.
-</MathBox>
-
 <MathBox title='Field' boxType='definition'>
 A field $\mathbb{F}$ is a set containing $1 \neq 0$ equipped with the binary operations $+$ (addition) and $\cdot$ (multiplication) such that
 - $(\mathbb{F}, +)$ is an abelian group
 - $(\mathbb{F}\setminus\Set{0}, \cdot)$ is an abelian group
-- Multiplication is distributive related to addition, i.e. for all $a,b,c\in\mathbb{F}$
-  - $a(b + c) = ab + ac$
-  - $(a + b)c = ac + bc$
+- Addition and multiplication are related by distributivity, i.e. for all $a,b,c\in\mathbb{F}$
+    - $a(b + c) = ab + ac$
+    - $(a + b)c = ac + bc$
 </MathBox>
 
 <MathBox title='Algebra' boxType='definition'>
@@ -289,4 +278,60 @@ for which the following properties hold:
 - $A$ is a vector space over $\mathbb{F}$ under addition and scalar multiplication
 - $A$ is a ring under addition and multiplication
 - if $r\in\mathbb{F}$ and $a,b\in A$ then $r(ab) = (ra)b = a(rb)$ (scalar compatibility)
-</MathBox>
+</MathBox>
+
+# Ring theory
+
+<MathBox title='Ring' boxType='definition'>
+A ring is a non-empty set $R$ equipped with two binary (closed) operations, 
+- $+: R \times R \ni (a,b)\mapsto a + b \in R$ **(addition)**
+- $\cdot: R\times R \ni (a,b)\mapsto ab \in R$ **(multiplication)***
+
+with the properties
+- $(R, +)$ is an abelian group
+- $(R, \cdot)$ is a monoid
+- Addition and multiplication are related by distributity, i.e. for all $a,b,c\in R$
+    - $(a + b)c = ac + bc$
+    - $c(a + b) = ca + cb$
+
+A ring is commutative if $ab = ba$ for all $a,b\in R$. If a ring $R$ contains an element $e$ such that $ae = ea = a$ for all $a\in R$, we say that $R$ is a ring with identity.
+</MathBox>
+
+<MathBox title='Zero divisors and units' boxType='definition'>
+Let $a,b$ be in a ring $R$. If $a \neq 0$ and $b \neq 0$ but $ab = 0$, then $a$ and $b$ are *zero divisiors*. If $ab = ba = 1$, we say the $a$ is a *unit*, or that $a$ is invertible.
+</MathBox>
+
+<MathBox title='Integral domain, division ring (skew field) and field' boxType='definition'>
+An *integral domain* is a commutative ring with no zero divisor. A *division ring* or *skew field* is a ring in which every non-zero element $a$ has an inverse $a^{-1}$. A *field* is a commutative ring in which every non-zero element is invertible.
+</MathBox>
+
+<MathBox title='Ring characteristic' boxType='definition'>
+The *characteristic* of a ring $R$, denoted $\operatorname{char}(R)$, is the smallest positive integer such that
+
+$$
+  n\cdot 1 = \underbrace{1 + \cdots + 1}_{n \text{ times}} = 0
+$$
+
+If there is no such positive integer, then $\operatorname{char}(R) = 0$.
+</MathBox>
+
+<MathBox title='Subring' boxType='definition'>
+A subring of a ring $R$ is a subset $S$ of $R$ forming a ring under the operations of addition and multiplication defined in $R$.
+</MathBox>
+
+<MathBox title='Ring homomorphism' boxType='definition'>
+Let $R, S$ be two rings. A map $f:R\to S$ is a ring homomorphism if it satisfies for $a,b\in R$
+1. $f(a + b) = f(a) + f(b)$ (group homomorphism)
+2. $f(ab) = f(a)f(b)$
+3. $f(1_R) = 1_S$
+</MathBox>
+
+<MathBox title='Ideals' boxType='definition'>
+Let $\mathcal{I}$ be a subset of ring $R$. An additive subgroup of $R$ is called
+- a *left ideal* of $R$ if $ra\in\mathcal{I}$ for $a\in\mathcal{I}$ and $r\in R$
+- a *right ideal* of $R$ if $ar\in\mathcal{I}$ for $a\in\mathcal{I}$ and $r\in R$
+
+A left ideal that is also a right ideal, is called a *(two-sided) ideal* of $R$. An ideal is *proper* if $\mathcal{I} \neq R$, and it is *non-trivial* if in addition $\mathcal{I}\neq 0$.
+</MathBox>
+
+<MathBox title='Proper and non-trivial ideal' boxType='definition'>