Let $r\in R$ and let $B$ be any $R$-submodule of a right $R$-module $A$. Then $A/(Ar+B)\cong A/Ar$.
In the proof, I have defined the map $f:A\to A/Ar$ by $f(a)=a+Ar$ for all $a\in A$. $f$ is well defined and is an $R$-module homomorphism. $f$ is onto. However, $\ker(f)=\{a\in A \mid a+Ar=Ar\}=\{a \mid a\in Ar\}=Ar$. I am failing to show that $A/(Ar+B)\cong A/Ar$. May be it is not true in general!!