Let $R$ be a domain and $a,b$ $\in$ $R$, prove $R/Rb$ $\cong_R$ $Ra/Rab$

Question

I was trying to prove the statement in the title, but couldn't get far. I tried creating a homomorphism from $R/Rb$ $\longrightarrow$ $Ra/Rab$ and prove is it isomorphic, failed at that. Looked for this exercise online and found a book that has this same exercise in which they gave a hint : "Use second and third isomorphism theorems". Still this does not help me, can anyone give me a hint as to what direction I have to look for ? The "closest" I have got to the statement : $"Ra/Rab \cong (Ra + Rb)/Rb"$.

Thanks for reading my question.

Answer 1

Use homomorphism from R to Ra/Rab which sends x to ax+Rab and it is easy to show that its kernnel is Rb