Let Z* denote the ring of integers with new addition and multiplication operatio

Question

Let Z* denote the ring of integers with new addition and multiplication operations defined by a (+) b = a + b - 1 and a (*) b = a + b - ab. Prove Z (the integers) are isomorphic to Z*.

Note: Please provide your observation of how you found the function that correctly maps these two rings. Thanks :)

Explanation / Answer

we are required to show f(a+b) = f(a) (+)f(b) and f(a.b)= f(a)(*)f(b), f is one one , f is onto. keeping the requirements in view, we define f:Z --> Z* by f(x)=-(1-x) then x= y -(1-x) = -(1-y) f(x)=f(y) so, f is well defined and one to one. suppose z is any integer in the codomain. then z + 1 is a integer in the domain ( which is also the set of integers) such that f(z+1) = z+1-1= z. that means every z in the codomain has a preimage in the domain under f. so, f is onto. now, consider f(x+y) = -(1-(x+y)) by definition of f. = x+y-1 = f(x)(+)f(y) by definition of (+). also, f(x.y) = -(1-x.y)) = -(1-(-x+x-xy-y+y+1)) =-(- (1-x).-(1-y)) =f(x)*f(y) therefore f preserves operations . thus, f is a ring isomorphism from (z,+,.) to (z,(+),(*)).

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