Thread: is an ideal of a subring of a ring?

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

Originally Posted by ziggychick

Is N necessarily a subring of R?

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).

thanks. that was very helpful.

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