From fa1e6f75ceb4f2a99dc1c107bfe60434356cfe19 Mon Sep 17 00:00:00 2001
From: JJ
Date: Tue, 20 Feb 2024 17:28:49 -0800
Subject: meow

---
 mathematics/algebra.md | 30 +++++++++++-------------------
 1 file changed, 11 insertions(+), 19 deletions(-)

(limited to 'mathematics/algebra.md')

diff --git a/mathematics/algebra.md b/mathematics/algebra.md
index 4b9a097..3756604 100644
--- a/mathematics/algebra.md
+++ b/mathematics/algebra.md
@@ -5,8 +5,8 @@ title: mathematics/algebra
 
 # algebra
 
-modern algebra is the study of **algebraic structures**: groups, rings, fields, modules, vector spaces, lattices, and the like.
-these structures are very general: and so results from abstract algebra can be applied to a wide variety of situations.
+modern algebra is the study of **algebraic structures**: groups, rings, fields, modules, and the like.
+these structures are very abstract: and so results can be applied to a wide variety of situations.
 
 ## structures
 
@@ -34,9 +34,9 @@ A **ring** $R$ is a set with two binary operations $+$ and $×$ satisfying the f
   - $∀a,b,c : (a+b) × c = (a×c)+(b×c)$
   - $∀a,b,c : a × (b+c) = (a×b) + (a×c)$
 - An Abelian or **commutative ring** satisfies an additional axiom:
-  - commutativity (of ×): $∀a,b : a×b=b×a$
+  - commutativity (of $×$): $∀a,b : a×b=b×a$
 
-A **field** is a *commutative ring* where $0 ≠ 1$ and all elements sans $0$ have an inverse $a^{-1}$ under multiplication.
+A **field** is a *commutative ring* where all elements sans $0$ have an inverse $a^{-1}$ under multiplication. Subsequently, $0 ≠ 1$. A field may be also thought of as a set on which addition, subtraction, multiplication, and division are defined and behave as they do on $ℝ$.
 
 A **vector space** $V$ over a field $F$ of scalars is a set with a binary operation $+$ and a binary function satisfying the following axioms:
 - $(V, +)$ is a *commutative group*:
@@ -53,7 +53,7 @@ A **vector space** $V$ over a field $F$ of scalars is a set with a binary operat
 
 A **module** $M$ is a generalization of a *vector space* to function over a ring $R$ instead of a field.
 
-A **lattice** $L$ is a set with two binary operations ∧ and ∨ satisfying the following axioms:
+A **lattice** $L$ is a set with two binary operations $∧$ and $∨$ satisfying the following axioms:
 - commutativity:
   - $∀a,b : a ∧ b = b ∧ a$
   - $∀a,b : a ∨ b = b ∨ a$
@@ -67,22 +67,14 @@ A **lattice** $L$ is a set with two binary operations ∧ and ∨ satisfying the
   - $∀a : a ∧ a = a$
   - $∀a : a ∨ a = a$
 
-## group theory
+## [group theory](group-theory.md)
 
-...
+## [ring theory](ring-theory.md)
 
-## ring theory
+## [galois theory](galois-theory.md)
 
-...
+## [linear algebra](linear-algebra.md)
 
-## galois theory
+## [order theory](order-theory.md)
 
-...
-
-## linear algebra
-
-...
-
-## order theory
-
-a lattice may alternatively be defined as...
+## [coding theory](coding-theory.md)
-- 
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