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

Thanks

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- May 11th 2009, 11:15 PMsidiideals
I have one problem with ideals. Which properties of ideal will cause that the factor ring is comutative or ring with 1?

Thanks - May 11th 2009, 11:51 PMNonCommAlg
first $\displaystyle I$ has to be a two sided ideal because otherwise the factor ring $\displaystyle R/I$ wouldn't be defined. now commutativity of $\displaystyle R/I$ means that $\displaystyle (x+I)(y+I)=(y+I)(x+I),$ for all $\displaystyle x,y \in R,$ i.e.

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

or equivalently $\displaystyle rx - x \in I$ and $\displaystyle xr - x \in I.$ you can't put this condition in a more familiar form! - May 12th 2009, 02:06 AMadit38
great......, thx bro