Isomorphism of Rings
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 :))
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)