Let R and S be rings and f : R→ S be a homomorphism. If x is an idempotent in R,
show that f(x) is an idempotent in S. Hence, or otherwise, determine all ring
homorphisms from Z×Z to Z .
Since x is idempotent in R, we have that
Consider,
Hence f(x) is idempotent in S.
The idempotent of are (0,0), (0,1), (1,0) and (1,1)
Aside from the zero morphism and identity morphism, we seek to see if there are others.
To do this consider,
(1,1) = (0,1) +(1,0)
Also, it is expected that f(1,1) =1 since f is a ring homomorphism.
So, we have 1 = f(1,1) = f ((0,1)+ (1,0)) = f(0,1) + f(1,0)
Case 1
If f(0,1) = 0 then f(1,0) = 1
And we must have
f(a,b) = f(a(0,1) + b(1,0)) = a f(0,1) + b f(1,0) = b
i.e f(a,b) = b
Case 2
If f(0,1) = 1 then f(1,0) = 0
And we must have
f(a,b) = f(a(0,1) + b(1,0)) = a f(0,1) + b f(1,0) = a
i.e f(a,b) = a
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