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?
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).