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Math Help - ring theory

  1. #1
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    ring theory

    Question:
    Let R be a ring. Show that R = 0 = {0} if and only if 1 = 0 in R

    How do you show it? I just know that 1 is the identity of multiplication group and 0 is the identity of addition group.

    Thanks alot
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  2. #2
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    Quote Originally Posted by knguyen2005 View Post
    Question:
    Let R be a ring. Show that R = 0 = {0} if and only if 1 = 0 in R

    How do you show it? I just know that 1 is the identity of multiplication group and 0 is the identity of addition group.

    Thanks alot
    IF 1=0
    Consider any x in R. x=1.x=0.x=0
    Thus R = {0}

    If R={0} not sure if there is anything to prove. x.0 = 0.x = x for all x in R. Thus by definition 1 = 0.
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  3. #3
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    Thanks for your reply, I understand the 1st part but I am not sure about last part
    you said x.0 = 0.x = x for all x in R.
    Can you explain to me how do you get this?

    Thanks again
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  4. #4
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    Quote Originally Posted by knguyen2005 View Post
    Thanks for your reply, I understand the 1st part but I am not sure about last part
    you said x.0 = 0.x = x for all x in R.
    Can you explain to me how do you get this?

    Thanks again
    Hi -
    R = {0} so when I say all x in R, only value x can take is x=0. I said for all x in R just to make it exactly similar to how you define element 1 for any Ring.
    if e.x = x.e = x for ALL x in R, then e is unique and is called multiplicative unity (1).

    Now in our example R={0), put e=0
    0.x = x.0 = x for all in x R.

    For any other Ring this is true ONLY if x=0. But as in our special example x=0 is the only element in the Ring we can use for all x in R and hence the definition of 1 is satisfied, thus e=0=1.

    I beleive this question is more to do with the understading of definitions than anything else.
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