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Math Help - Isomorphism of Rings

  1. #1
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    Isomorphism of Rings

    Hello,
    I was just trying out problems from Herstein, and I have a doubt in this. I would really appreciate any help.

    Let R be a ring with unit element. Using its elements we define a ring R' by defining a (+) b = a+b+1 and a.b=a+b+ab where a,b are in R and where the addition and multiplication on the right hand side of these relations are those of R.
    a) Prove that R' is a ring under (+) and .
    This I was able to do.

    b) What acts as the zero-element of R'?
    The zero-elt of R' is -1

    c) What acts as the unit element of R'?
    The unit elt of R' is 0.

    d) Prove that R is isomorphic to R'.
    This is where I am having problems. I need to define a map f:R' --> R which is 1-1, onto and a homomorphism.
    I can also see that f shouls map the zero elts (ie) -1~~> 0 and since f should be onto, the unit element to the unit elt of R (ie) 0~~>1

    The only map I can think of is f(a) = a+1

    Using that f(a(+)b) = f(a + b + 1) = a + b + 1 +1 = a +1 + b +1 = f(a) + f(b)
    f(a.b) = f(a + b + ab) = a+b +ab + 1 = . this should be ab to make it f(a)f(b). Where have I gone wrong, is the map itself wrong??

    Thank you )
    Last edited by poorna; May 12th 2010 at 09:32 PM.
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  2. #2
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    Oh i cant believe i missied something simple,
    f(a.b) = f(a +b +ab) = a+ b+ ab +1 = (a +1) +b(a+1) =(a+1)(b+1) = f(a)f(b)
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