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Math Help - Algebra.....units prove

  1. #1
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    Red face Algebra.....units prove

    Suppose that R and S are rings with identity. Let (r, s) in R S. Show that
    (r, s) is unit in R S <=> r is a unit in R and s is a unit in S.

    Also, if (r, s) is a unit in RS, show that inverse of (r, s) = (inverse of r, inverse of s)
    (Hint: It is efficient to prove the last statement and the implication <= in the first statement at the
    same time.)


    um....can anyone help!!! urgent!!! Thanks very much
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  2. #2
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    Quote Originally Posted by suedenation
    Suppose that R and S are rings with identity. Let (r, s) in R S. Show that
    (r, s) is unit in R S <=> r is a unit in R and s is a unit in S.

    Also, if (r, s) is a unit in RS, show that inverse of (r, s) = (inverse of r, inverse of s)
    (Hint: It is efficient to prove the last statement and the implication <= in the first statement at the
    same time.)


    um....can anyone help!!! urgent!!! Thanks very much
    If S,R are rings with unity, then their direct product is also a ring S\times R. Let (s,r)\in S\times R. Then, s\in S and r\in R. If (s,r) is a unit there exists, (s^{-1},r^{-1}) but s^{-1}\in S and r^{-1}\in R. Now assume the converse, if s\in S and r\in R are units, then there exists s^{-1}\in S and r^{-1}\in R, thus, (s^{-1},r^{-1})\in R\times S is a unit.
    Q.E.D.
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