commutative ring

Let (X,+,) be a ring. Since (X,+) is required to be an abelian groupMathworldPlanetmath, the operationMathworldPlanetmath+” necessarily is commutativePlanetmathPlanetmath.

This needs not to happen for “”. Rings R where “” is commutative, that is, xy=yx for all x,yR, 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 inverseMathworldPlanetmath 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 divisorsMathworldPlanetmath, i.e. nonzero elements having productPlanetmathPlanetmath zero. Since the ideals of a commutative ring are two-sided (, 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