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).