[Math] Ring isomorphism, Unity is preserved

Question

I am wondering that if I have a ring isomorphism, /phi going from R to R', where R is a ring with unity, how can I prove that R' is also a ring with unity? It seems to be very obvious so I don't know where to start…. Just tell me something to keep me going thanks!!

Answer 1

Yes. If $\phi:R\to S$ is a ring homomorphism, then $$ϕ(r)=ϕ(1\cdot r)=ϕ(1)\cdotϕ(r)$$ so $ϕ(1)$ is a unity with respect to $ϕ[R]$, which is a subring of $S$. So if $ϕ$ is surjective, then $ϕ(1)$ is the unity of $S$.