commutative ring

Let $(X,+,\cdot )$ be a ring. Since $(X,+)$ is required to be anabelian 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 fieldsthan other rings are, but there are certain dissimilarities. Afield has always a multiplicative inverse for each of itsnonzero elements, but the same needs not to be true for acommutative ring. Further, in a commutative ring there mayexist zero divisors, i.e. nonzero elements having product zero.Since the ideals of a commutative ring aretwo-sided (http://planetmath.org/Ideal), thethese rings are more comfortable to handle than other rings.

The study of commutative rings is called commutative algebra.