---
layout: algebra
title: mathematics/algebra
---

# algebra

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

An **algebraic structure** is a set with a collection of *operations* and a finite set of *axioms* those operations must satisfy.

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: $∀a, ∃a^{-1} : a⋆a^{-1} = e$
- An Abelian or **commutative group** satisfies an additional axiom:
  - commutativity: $∀a,b : a⋆b=b⋆a$

A **monoid** is a group without an inverse operation.

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: $∃0, ∀a : 0+a = a+0 = a$
  - additive inverse: $∀a, ∃-a : a+(-a) = 0$
  - commutativity: $∀a,b : a+b=b+a$
- $(R, ×)$ is a *monoid*
  - associativity: $∀a,b,c : (a×b)×c = a×(b×c)$
  - multiplicative identity: $∃1, ∀a : 1×a = a×1 = a$
- The *distributive laws* hold for + and ×:
  - $∀a,b,c : (a+b) × c = (a×c)+(b×c)$
  - $∀a,b,c : a × (b+c) = (a×b) + (a×c)$
- An Abelian or **commutative ring** satisfies an additional axiom:
  - commutativity (of $×$): $∀a,b : a×b=b×a$

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**](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: $∃0, ∀v: 0+v = v+0 = v$
  - additive inverse: $∀v, ∃-v: v+(-v) = 0$
  - commutativity: $∀u,v : u+v=v+u$
- $(V, )$ is a *scalar operation*:
  - scalar identity: $∃1 ∈ F, ∀v ∈ V : 1v = v1 = v$
  - commutativity: $∀a,b ∈ F, ∀v ∈ V : (ab)v = a(bv)$
- The *distributive laws* hold:
  - $∀a ∈ F, ∀u,v ∈ V : a(u+v) = au+av$
  - $∀a,b ∈ F, ∀v ∈ V : (a+b)v = av + bv$

A **module** $M$ is a generalization of a *vector space* to function over a ring $R$ instead of a field.

A [**lattice**](order-theory) $L$ is a set with two binary operations $∧$ and $∨$ satisfying the following axioms:
- commutativity:
  - $∀a,b : a ∧ b = b ∧ a$
  - $∀a,b : a ∨ b = b ∨ a$
- associativity:
  - $∀a,b,c : a ∧ (b ∧ c) = (a ∧ b) ∧ c$
  - $∀a,b,c : a ∨ (b ∨ c) = (a ∨ b) ∨ c$
- absorption:
  - $∀a,b : a ∧ (a ∨ b) = a$
  - $∀a,b : a ∨ (a ∧ b) = a$
- idempotence:
  - $∀a : a ∧ a = a$
  - $∀a : a ∨ a = a$

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).