I was wondering whether for a ring $R$ and two elements $a,b$ that $R/(a,b)$ is isomorphic to $(R/(a))/(b)$ (with the parenthesis representing the smallest ideal containing the respective element/s).
I have seen this trick applied to polynomial rings on multiple occations.
Thank you
The Third Isomorphism Theorem tells you that for a ring $R$, and ideals $I\leq K\leq R$, you have that $K/I$ is an ideal of $R/I$ and that $$\frac{R/I}{K/I}\cong \frac{R}{K}.$$ We can apply this to you situation by taking $I=(b)$, $J=(a)$, and $K=I+J=(a,b)$.
Note that the smallest ideal containing $b+(a)$ in $R/(a)$ is in fact the ideal $(b)+(a)=(a,b)+(a)=K/(a)$. Thus, you are asking whether $\frac{R/(a)}{K/(a)}$ is isomorphic to $R/K$. And then the Third Isomorphism Theorem tells you immediately that the answer is “yes”.