Hi,

I'm a little confused about ideals.

In our text an ideal is defined as:
For a ring R, an additive subgroup of ring were for each aA and nN, and are both in N.

Is N necessarily a subring of R?

Thanks

It depends how you define subrings. If you are a commutative algebraist then you usually think of rings as commutative unitary rings and subrings consist of 0 and 1 which are subsets and satisfy the properties of a ring under the induced operations. If you are a noncommutative algebraist then you usually think of rings are as general rings (which might or might not have unity) and define a subring to be a subset which satifies the properties of a ring under the induced operations. If you are a blondie then you define rings as what your boyfriend buys for you on Christmans (just joking if you happen to be a blonde ).

Under the more general definition ideals happen to be subrings, however, under the commutative algebraist definition ideals are not subrings unless they contain 1 (but in that case then they happen to be the entire ring themselves).

__________________

To view links or images in signatures your post count must be 10 or greater. You currently have 0 posts.


"Democracy has proved only that the best way to gain power
over people is to assure the people that they are ruling
themselves. Once they believe that, they make wonderfully
submissive slaves." - Joseph Sobran

thanks. that was very helpful.

and i'm not a blondie but if i do get a ring for christmas i hope it commutes.