# Ideal

• Nov 11th 2006, 06:21 PM
JaysFan31
Ideal
Let R be any commutative ring and let a1,......,an be elements of R. We define
(a1,.....,an) to be the set {r1a1+...rnan for all i, ri in R}.
Prove that (a1,......,an) is an ideal of R. We call it the ideal generated by a1,......, an.

I appreciate any help.
• Nov 11th 2006, 06:35 PM
ThePerfectHacker
Quote:

Originally Posted by JaysFan31
Let R be any commutative ring and let a1,......,an be elements of R. We define
(a1,.....,an) to be the set {r1a1+...rnan for all i, ri in R}.
Prove that (a1,......,an) is an ideal of R. We call it the ideal generated by a1,......, an.

Let us show that,
$\displaystyle S=\{r_1a_1+...+r_na_n\}$
Has property that,
$\displaystyle xS\subseteq S, Sx\subseteq S$
Well that is simple to show,
Any element of $\displaystyle xS$ can be expressed as,
$\displaystyle \{xr_1a_1+...+xr_na_n\}$
And every $\displaystyle xr_i\in R$ (for it is closed since it is a ring).

Now the other way around $\displaystyle Sx\subseteq S$ is also true for any element of $\displaystyle Sx$ can be expressed as,
$\displaystyle r_1a_1x+...+r_na_nx$
But since $\displaystyle R$ is a commutative ring,
$\displaystyle xr_1a_1+...+xr_na_n$.
That part is finished.

The question remains to show that this generating set is an additive subgroup of the ring. I am not going to show that (in fact it is simple to) but you show be familar from group theory that these generating sets always forms a subgroup (if one element only then it is a cyclic subgroup).