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