Commutative Ring

Idea

A ring is a algebraic generalization of the integers.

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:

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

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 Group or 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

• All fields are rings where every non-zero element has a multiplicative inverse. Examples include:
• The set of integers $\mathbb{Z}$ forms a prototypical commutative ring under standard addition and multiplication.
• The collection of Polynomials of a Commutative Ring forms a ring structure, typically denoted $R[x]$.
• The set of Matrices over a Ring of size $n\times n$ itself forms a ring. These are non-commutative whenever $n>1$.

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

Properties

• Every commutative ring has a unique multiplicative identity (if it is a unital ring)
• The ideals of a commutative ring form a lattice under inclusion
• Prime and maximal ideals play a central role in commutative algebra
• Commutative rings admit localizations

Examples

• The integers $\mathbb{Z}$ with standard addition and multiplication
• Polynomial rings $R[x]$ over a commutative ring $R$
• Fields are commutative rings where every non-zero element has a multiplicative inverse
• The ring $\mathbb{Z}/n\mathbb{Z}$ of integers modulo $n$

Related Concepts

• Ring: The more general structure
• Field: A commutative ring where every non-zero element is invertible
• Module: Generalizes vector spaces over commutative rings
• Ideal: Substructures of rings
• Polynomial Ring: Constructed from commutative rings