Ring

Definition

A ring is an algebraic structure $(A,+,\cdot )$ consisting of a set $A$ equipped with two binary operations, addition and multiplication. Their signature is:

$$\begin{array}{rl}A& :\star \\ +& :A\to A\to A\\ \cdot & :A\to A\to A\end{array}$$

A ring must satisfy the following properties:

• $(A,+)$ forms an Abelian group.
• $(A,\cdot )$ forms a monoid (possessing an identity element and satisfying associativity).
• Multiplication distributes over addition: $a\cdot (b+c)=(a\cdot b)+(a\cdot c)$ and $(a+b)\cdot c=(a\cdot c)+(b\cdot c)$ for all $a,b,c\in A$.

Variants

• Rngs: Some definitions exclude the requirement for a multiplicative identity, in which case $(A,\cdot )$ may be viewed as a semigroup or a Magma with associativity.
• Commutative Ring: A ring in which the multiplication operation $\cdot$ is commutative, such that $a\cdot b=b\cdot a$ for all $a,b\in A$.

Examples

Fields

All fields are rings where every non-zero element has a multiplicative inverse. Examples include:

• The rational numbers $\mathbb{Q}$
• The real numbers $\mathbb{R}$
• The complex numbers $\mathbb{C}$

Integers

The set of integers $\mathbb{Z}$ forms a prototypical commutative ring under standard addition and multiplication.

Polynomials

The collection of Polynomials of a Commutative Ring forms a ring structure, typically denoted $R[x]$.

Matrices

The set of Matrices over a Ring of size $n\times n$ forms a ring. This is a primary example of a non-commutative ring when $n>1$.

Function Rings

The Ring of Functions from a Set to a Ring is a ring where operations are defined pointwise.

Related Concepts

• Abelian Group: The additive structure of a ring
• Monoid (Category Theory): The multiplicative structure of a ring
• Ideal: A sub-structure used to construct quotient rings
• Ring Homomorphism: A map preserving the ring structure