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 (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