# commutative ring

Let $(X,+,\cdot )$ be a ring. Since $(X,+)$ is required to be an
abelian group^{}, the operation^{} “$+$” necessarily is commutative^{}.

This needs not to happen for “$\cdot $”. Rings $R$ where “$\cdot $” is commutative, that is, $x\cdot y=y\cdot x$ for all $x,y\in R$, are called commutative rings.

The commutative rings are rings which are more like the fields
than other rings are, but there are certain dissimilarities. A
field has always a multiplicative inverse^{} for each of its
nonzero elements, but the same needs not to be true for a
commutative ring. Further, in a commutative ring there may
exist zero divisors^{}, i.e. nonzero elements having product^{} zero.
Since the ideals of a commutative ring are
two-sided (http://planetmath.org/Ideal), the
these rings are more comfortable to handle than other rings.

The study of commutative rings is called commutative algebra.

Title | commutative ring |
---|---|

Canonical name | CommutativeRing |

Date of creation | 2013-11-12 18:26:58 |

Last modified on | 2013-11-12 18:26:58 |

Owner | drini (3) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | drini (2872) |

Entry type | Definition |

Classification | msc 13A99 |

Related topic | GroupOfUnits |

Related topic | ExampleOfRings |