Let $R$ be a ring and $A$ be a right ideal in $R$. Define $I_R(A) = \{r \in R | rA \subseteq A \}$. If $B$ be an ideal in $R$ and $B \subseteq A$, find all elements in $I_{R/B}(A/B)$ and $(I_R(A))/B$.
My attempt:
I found that \begin{align*} I_{R/B}(A/B) &= \{m \in R/B | m(A/B) \subseteq (A/B) \} \\ &= \{r \in R | (r+B)(a+B) \subseteq A/B, a \in A \} \\ &= \{r \in R | ra + B \subseteq A/B \subseteq A, a \in A \} \end{align*}
and
\begin{align*} (I_R(A))/B &= \{x \in I_R(A) | x+B \subseteq A \} \\ &= \{r \in R | rA + B \subseteq A \} \\ &= \{r \in R | ra + B \subseteq A, a \in A \} \end{align*}
Is that true? Please correct me if there are some mistakes. Thanks in advanced.