Def/Integral domain

Logical Connections

This definition logically relies on the following definitions and statements: Def/Commutative, Def/Ring, Def/Zero divisor

The following statements and definitions logically rely on the material of this page: Def/Euclidean domain, Def/Euclidean valuation, Def/Factorization into irreducibles, Def/Irreducible element, Def/PID, Def/Prime element, Def/UFD, Def/Unique factorization, State/Additivity of polynomial degrees, State/Divisibility corresponds to containment of principal ideals, and State/Fields are integral domains

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Basic ring theory

Suppose that $R$ is a commutative ring. Then $R$ is an integral domain, if and only if the following cancellation principle holds:

• If $x,y,z \in R$, and $xy = xz$, then $y = z$ or $x = 0$.

This article has been written by: User:Marty