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Math Help - rings and fields..urgent help plz

  1. #1
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    rings and fields..urgent help plz

    Hi! I have a midterm tomorrow and I am stuck on this problem. Can anyone help me??? Thankkk youu soo much.



    1) Suppose F is a field, and a is a nonzero element of F. Show that if r, s are in F and ar = as, then r = s.


    2) Let R be a ring with identity. Prove that for all a, b, c in R, if a + b = 0 and a + c = 0 then b = c. (This means that for any a in R, the element -a is uniquely determined.
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  2. #2
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    Quote Originally Posted by jenjen View Post
    1) Suppose F is a field, and a is a nonzero element of F. Show that if r, s are in F and ar = as, then r = s.
    Each non-zero element in a field has an inverse (by definition).
    Thus,
    a^{-1}(ar)=a^{-1}(as)
    Associate property,
    (a^{-1}a)r=(a^{-1}a)s
    1r=1s
    r=s

    2) Let R be a ring with identity. Prove that for all a, b, c in R, if a + b = 0 and a + c = 0 then b = c. (This means that for any a in R, the element -a is uniquely determined.
    What does it mean a ring with identity ???

    a+b=a+c
    Then,
    -a+(a+b)=-a+(a+c)
    (-a+a)+b=(-a+a)+c
    0+b=0+c
    b=c
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  3. #3
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    Thank you for the quick reply ThePerfectHacker!!

    Def: An element a of a ring with with identity R is called a unit of R if there exists some b in R so that a x b = b x a = 1
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  4. #4
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    I also found this defintion

    Def: A ring (with identity) is a set R with two operations, addition and mutiplication, and two special elements, 0 and 1, which satisfy axioms (associativity of addition, communtativity of addition, 0 is a zero element, etc.). The operations addition and multiplication may each be thought of as functions from R x R (ordered pairs of elements of the set R) to R, so that for any ordered pair (a,b), where a, b are in R, a + b is an element of R, and a x b is an element of R.
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  5. #5
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    Quote Originally Posted by jenjen View Post
    I also found this defintion

    Def: A ring (with identity) is a set R with two operations, addition and mutiplication, and two special elements, 0 and 1, which satisfy axioms (associativity of addition, communtativity of addition, 0 is a zero element, etc.). The operations addition and multiplication may each be thought of as functions from R x R (ordered pairs of elements of the set R) to R, so that for any ordered pair (a,b), where a, b are in R, a + b is an element of R, and a x b is an element of R.
    The standard terminology is a "ring with unity".
    Perhaps you are using a text from the begining from the 20th Centrury
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