Abstract Algebra · Hierarchy

Groups, rings, and fields are sets with operations that obey some of the rules of arithmetic. A group has one operation, which is associative, has an identity element, and can be undone. A ring has an addition and a multiplication; its addition forms a commutative group, but its elements need not have multiplicative inverses. A field is a commutative ring in which every nonzero element has a multiplicative inverse, so division by nonzero elements is possible.

field ⊂ ring ⊂ commutative group (under addition)

Definitions

group

Undergraduate (abstract algebra)

Consider a set $G$ with a binary operation $\ast$. The set $G$ is a group if

  1. $\ast$ is associative: $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a, b, c \in G$,
  2. $G$ has an identity element $e$ such that $e \ast g = g \ast e = g$ for all $g \in G$, and
  3. every element $g \in G$ has an inverse $g^{-1}$ such that $g \ast g^{-1} = g^{-1} \ast g = e$.
Examples
Non-examples
  • $\nats$ with addition
    $\nats$ with addition is not a group, because no positive integer has an additive inverse in $\nats$.
Properties
  • The identity element of a group is unique, and so is the inverse of each element.

ring

Undergraduate (abstract algebra)

Consider a set $R$ with two binary operations, addition $+$ and multiplication $\cdot$. The set $R$ is a ring if

  1. $R$ with $+$ is a commutative group, whose identity element is written $0$,
  2. multiplication is associative and has an identity element, written $1$, and
  3. multiplication distributes over addition: $a (b + c) = a b + a c$ and $(a + b) c = a c + b c$ for all $a, b, c \in R$.
Examples
Non-examples
Properties
  • Multiplication in a ring need not be commutative. A ring whose multiplication is commutative is called a commutative ring.
  • Elements of a ring need not have multiplicative inverses, so division need not be possible.
  • In every ring, $0 \cdot a = a \cdot 0 = 0$ and $(-1) \cdot a = -a$ for all $a$.

field

High school

A field is a set of numbers (or other objects) that can be added, subtracted, multiplied, and divided by every nonzero element, following the same rules as the arithmetic of real numbers.

Undergraduate (abstract algebra)

Consider a set $F$ with two binary operations, addition $+$ and multiplication $\cdot$. The set $F$ is a field if

  1. $F$ is a ring with $+$ and $\cdot$,
  2. multiplication is commutative,
  3. There exists an additive identity $0$ and a multiplicative identity $1$ such that $0 \neq 1$, and
  4. every element $a \neq 0$ has a multiplicative inverse $a^{-1}$ such that $a a^{-1} = 1$.
Examples
Non-examples
Properties
  • A field has no zero divisors: if $a b = 0$, then $a = 0$ or $b = 0$.
  • The integers modulo $n$ form a field if and only if $n$ is prime.
  • The scalars of a vector space come from a field.

Examples

Example group ring field Why
The nonzero real numbers, with multiplication ✓✗✗
The integers $\mathbb{Z}$, with addition and multiplication ✓✓✗
The rational numbers $\mathbb{Q}$, with addition and multiplication ✓✓✓
The $2 \times 2$ real matrices, with matrix addition, multiplication by real numbers, and matrix multiplication ✓✓✗
The integers modulo $6$, $\setdef{0, 1, 2, 3, 4, 5}$, with addition and multiplication modulo $6$ ✓✓✗
The integers modulo $5$, $\setdef{0, 1, 2, 3, 4}$, with addition and multiplication modulo $5$ ✓✓✓ The integers modulo $5$ form a field, but the integers modulo $6$ do not, because $5$ is prime and $6$ is not.

Notes

  • A ring is a group under addition, but a ring is usually not a group under multiplication, because $0$ has no multiplicative inverse.
  • The nonzero elements of a field are a group under multiplication.
  • Some authors do not require a ring to have a multiplicative identity $1$.

See also

All distinctions · All definitions