# ideals

• May 12th 2009, 12:15 AM
sidi
ideals
I have one problem with ideals. Which properties of ideal will cause that the factor ring is comutative or ring with 1?
Thanks
• May 12th 2009, 12:51 AM
NonCommAlg
Quote:

Originally Posted by sidi

I have one problem with ideals. Which properties of ideal will cause that the factor ring is comutative or ring with 1?
Thanks

first $I$ has to be a two sided ideal because otherwise the factor ring $R/I$ wouldn't be defined. now commutativity of $R/I$ means that $(x+I)(y+I)=(y+I)(x+I),$ for all $x,y \in R,$ i.e.

$xy-yx \in I.$ so $R/I$ is commutative iff $I$ contains all additive commutators. for $R/I$ to have 1, there must exist $r \in R$ such that $(r+I)(x+I)=(x+I)(r+I)=x+I,$ for all $x \in R,$

or equivalently $rx - x \in I$ and $xr - x \in I.$ you can't put this condition in a more familiar form!
• May 12th 2009, 03:06 AM