Commutative Ring

Definition

A commutative ring is a Ring $(R,+,\cdot ,0,1)$ in which multiplication is commutative: for all $a,b\in R$,

$a\cdot b=b\cdot a$

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
• CRing (Category): The category of commutative rings
• Module: Generalizes vector spaces over commutative rings
• Ideal: Substructures of rings
• Polynomial Ring: Constructed from commutative rings