Commutative Ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra.

Some specific kinds of commutative rings are given with the following chain of class inclusions:

Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields

Algebraic structures
Group-like structures Semigroup and Monoid Quasigroup and Loop Abelian group
Ring-like structures Semiring Near-ring Ring Commutative ring Integral domain Field
Lattice-like structures Semilattice Lattice Map of lattices
Module-like structures Group with operators Module Vector space
Algebra-like structures Algebra Associative algebra Non-associative algebra Graded algebra Bialgebra

Famous quotes containing the word ring:

“Ring out the old, ring in the new,
Ring happy bells, across the snow:
The year is going, let him go;
Ring out the false, ring in the true.”
—Alfred Tennyson (1809–1892)