Thread: 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.

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,

Has property that,

Well that is simple to show,
Any element of can be expressed as,

And every (for it is closed since it is a ring).

Now the other way around is also true for any element of can be expressed as,

But since is a commutative ring,
.
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).