Prove or disprove : Let I be an ideal of R and R/I be a integral domain then R is an integral domain.
I am confused this is true or not. My approach is if I assume that R is not a integral domain then there exist some a,b$\in$ R-I such that ab=0 then (a+I)(b+I)=ab+I=I,which contradicts the fact that R/I is a integral domain. But I am not sure if my proof is right or wrong. It will be enough if you give me some hint and if there is a better solution I will appreciate that. Thank you.
Your mistake is in assuming that if $R$ is not an integral domain then you must be able to find a pair of zero divisors, neither element living in $I$. You wrote $a, b \in R-I$, but there is no guarantee of that. Instead, all you can say is that $a, b \in R$ are both non-zero yet $ab =0$. This does not yield a contradiction once you carry out your coset multiplications.
Instead, the claim is false. You can quotient out $\{0,2\}$ from the ring $\mathbb{Z}_4$ (not an integral domain) and get (something isomorphic to) $\mathbb{Z}_2$, which is a field.