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 |
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Group-like structures
Semigroup and Monoid Quasigroup and Loop Abelian group |

Ring-like structures
Semiring Near-ring Ring Commutative ringIntegral 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 |

Read more about Commutative Ring: Ideals and The Spectrum, Ring Homomorphisms, Modules, Noetherian Rings, Dimension, Constructing Commutative Rings, Properties

### Famous quotes containing the word ring:

“These words dropped into my childish mind as if you should accidentally drop a *ring* into a deep well. I did not think of them much at the time, but there came a day in my life when the *ring* was fished up out of the well, good as new.”

—Harriet Beecher Stowe (1811–1896)

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