Ring with p elements

Let us prove that every (unitary) ring R with p elements, p being prime, is isomorphic to ℤp. Let us consider the homomorphism φ:ℤ—>R , 1|—>1. We have that Ker(φ) is an ideal of ℤ so Ker(φ)=nℤ for some n∈ℕ. Now by the first isomorphism theorem ℤ/Ker(φ) is isomorphic to Im(φ). By Lagrange’s theorem we have:

n=|ℤ/nℤ|=|ℤ/Ker(φ)|=|Im(φ)| | |R|= p

Since p is prime and n≠1 (since 1∉Ker(φ)) we have n=p. It follows that:

ℤ/pℤ = ℤ/Ker(φ) ≅ Im(φ)=R.