Docs: Added Commutative Ring.md, Division Ring.md and Field.md (#87)

* Docs: Added Commutative Ring.md, Division Ring.md and Field.md

In Basic Algebra, to describe field, added commutative ring and
division ring

* fix typo

* Added Commutative Ring and Division Ring docs to Basic Algebra.md

Author: minsubb13

content/Basic Algebra/Basic Algebra.md

@@ -11,6 +11,8 @@
 - [[Equivalence Relation]]
 - [[Quotient Ring]]
 - [[Ring Homomorphism]]
+- [[Commutative Ring]]
+- [[Division Ring]]
 - [[Field]]
 - [[Finite Field]]
 - [[Cyclotomic polynomial]]

content/Basic Algebra/Commutative Ring.md

@@ -0,0 +1,9 @@
+# Definition
+
+A ring in which the multiplication is commutative is a **commutative ring**. A ring with a multiplicative identity element is a **ring with unity;** the multiplicative identity element $1$ is called "**unity**".
+
+In a ring with unity $1$ the distributive laws show that
+
+$$(\underbrace{1 + 1 + \cdots + 1}_{n \text{ summands}})(\underbrace{1 + 1 + \cdots + 1}_{m \text{ summands}}) = (\underbrace{1 + 1 + \cdots + 1}_{nm \text{ summands}})$$
+
+that is, ${(n\cdot 1)}{(m\cdot 1)} = {(nm)\cdot 1}$

content/Basic Algebra/Division Ring.md

@@ -0,0 +1,3 @@
+# Definition
+
+Let $R$ be a ring with unity $1\neq 0$. An element $u$ in $R$ is a **unit** of $R$ if it has a multiplicative inverse in $R$. If every nonzero element of $R$ is a unit, then $R$ is a **division ring** (or **skew field**).

content/Basic Algebra/Field.md

@@ -0,0 +1,22 @@
+# Definition
+
+A field $F$ is a set equipped with two binary operations, addition ($+$) and multiplication ($\times$), that satisfy the following axioms:
+
+1. Additive Structure:
+    - $(F, +)$ forms an [[Abelian Group]].
+    - The additive identity is denoted as $0$
+
+2. Multiplicative Structure:
+    - $(F\{0\}, \times)$ forms an abelian group
+    - The multiplicative identity is denoted as $1$
+    - $1\neq 0$; the multiplicative identity ($1$) is not equal to the additive identity ($0$)
+
+3. Distributive Law:
+    - For all $a, b, c \in F: a \times (b + c) = (a \times b) + (a \times c)$
+
+More precisely, a field is a [[Commutative Ring|commutative]] [[Division Ring|division]] ring. In other words, a field is a [[Commutative Ring]] with unity where every nonzero element has a multiplicative inverse. A noncommutative division ring is called a "**strictly [[Division Ring|skew field]]**".
+
+## Example
+- The rational numbers ($\mathbb{Q}$)
+- The real numbers ($\mathbb{R}$)
+- The complex numbers ($\mathbb{C}$)