Let Φ: C⇒R be a ring homomorphism. Prove Φ(1)=0 or Φ(1)=1

C stands for complex numbers and R stands for real numbers

Suppose that Φ: C ⇒ R is a ring homomorphism.

Let Φ(1) = x for some x in R.

Then, since Φ is a ring homomorphism,
x = Φ(1) = Φ(1 * 1) = Φ(1) * Φ(1) = x * x.

Hence, x^2 = x
==> x(x - 1) = 0
==> x = 0 or 1.

I hope this helps!