meow

1 files changed, 5 insertions, 15 deletions

diff --git a/mathematics/algebra.md b/mathematics/algebra.md
index 0ba7847..3717445 100644
--- a/mathematics/algebra.md
+++ b/mathematics/algebra.md
@@ -11,7 +11,7 @@ Modern algebra is the study of **algebraic structures**: groups, rings, fields,
 
 An **algebraic structure** is a set with a collection of *operations* and a finite set of *axioms* those operations must satisfy.
 
-A **group** $G$ is a set with a single binary operation $⋆$ satisfying the following axioms:
+A [**group**](group-theory) $G$ is a set with a single binary operation $⋆$ satisfying the following axioms:
 - associativity: $∀a,b,c : (a⋆b)⋆c = a⋆(b⋆c)$
 - identity: $∃e, ∀a : e⋆a = a⋆e = a$
 - inverse: $∀a, ∃a^{-1} : a⋆a^{-1} = e$
@@ -20,7 +20,7 @@ A **group** $G$ is a set with a single binary operation $⋆$ satisfying the fol
 
 A **monoid** is a group without an inverse operation.
 
-A **ring** $R$ is a set with two binary operations $+$ and $×$ satisfying the following axioms:
+A [**ring**](ring-theory) $R$ is a set with two binary operations $+$ and $×$ satisfying the following axioms:
 - $(R, +)$ is a *commutative group*:
   - associativity: $∀a,b,c : (a+b)+c = a+(b+c)$
   - additive identity: $∃0, ∀a : 0+a = a+0 = a$
@@ -37,7 +37,7 @@ A **ring** $R$ is a set with two binary operations $+$ and $×$ satisfying the f
 
 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:
+A [**vector space**](linear-algebra) $V$ 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*:
   - associativity: $∀u,v,w : (u+v)+w = u+(v+w)$
   - additive identity: $∃0, ∀v: 0+v = v+0 = v$
@@ -52,7 +52,7 @@ 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**](order-theory) $L$ is a set with two binary operations $∧$ and $∨$ satisfying the following axioms:
 - commutativity:
   - $∀a,b : a ∧ b = b ∧ a$
   - $∀a,b : a ∨ b = b ∨ a$
@@ -66,14 +66,4 @@ A **lattice** $L$ is a set with two binary operations $∧$ and $∨$ satisfying
   - $∀a : a ∧ a = a$
   - $∀a : a ∨ a = a$
 
-## [group theory](group-theory)
-
-## [ring theory](ring-theory)
-
-## [galois theory](galois-theory)
-
-## [linear algebra](linear-algebra)
-
-## [order theory](order-theory)
-
-## [coding theory](coding-theory)
+An **algebra** $A$ over a field $F$ is a *vector space* equipped with an additional *bilinear product*. It is also common to consider algebras over a *ring* (and thus $A$ as a *module* with an additional product).