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Math Help - is an ideal of a subring of a ring?

  1. #1
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    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 \inA and n inN, a\dot n and n\dot a are both in N.

    Is N necessarily a subring of R?

    Thanks
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  2. #2
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    Quote Originally Posted by ziggychick View Post
    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).
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  3. #3
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    thanks. that was very helpful.

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