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 a A and n N, 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).