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Math Help - Rings, subrings problem

  1. #1
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    Rings, subrings problem

    Recall that an identity for a ring R is an element 1_R in R such that for each for each r in R,
    1_R*r = r = r*1_R
    (a) Show that there is a ring R with identity 1_R and a subring S of R not containing 1_R,
    but such that S has its own identity 1_S not equal to 1_R.
    (b) Show that if R is an integral domain then for every subring S with identity 1_S, 1_S = 1_R.

    Your help would be much appreciated. Thank you.
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  2. #2
    Ant
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    Re: Rings, subrings problem

    Quote Originally Posted by christianwos View Post
    Recall that an identity for a ring R is an element 1_R in R such that for each for each r in R,
    1_R*r = r = r*1_R
    (a) Show that there is a ring R with identity 1_R and a subring S of R not containing 1_R,
    but such that S has its own identity 1_S not equal to 1_R.
    (b) Show that if R is an integral domain then for every subring S with identity 1_S, 1_S = 1_R.

    Your help would be much appreciated. Thank you.
    For (a) consider \mathbb{Z} \oplus \mathbb{Z} then is (1,1) is the identity. Consider the subring  \{(a,0) : a \in \mathbb{Z} \} .....
    Last edited by Ant; April 21st 2013 at 04:42 PM.
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