Let $I$ and $J$ be ideals, $(I+J)/I\subseteq (I+J)$?

Question

Let $I$ and $J$ be ideal. We consider $$ \begin{split} (I+J)/I:=&\{(i+j)+I\;|\;i\in I,j\in J\}\\ =&\{j+(i+I)\;|\;i\in I,j\in J\}\\ =&\{j+I\;|\;j\in J\}. \end{split} $$ Can I conclude from this that $$(I+J)/I\subseteq (I+J).$$

Thanks in advance!

Answer 1

I think you need to be more careful with your definitions. The elements in $(I + J)/I$ are equivalence classes of elements in $(I + J)$.

We say that two elements in $(I + J)/I$ are equivalent if they differ by something which lives in $I$.

As such, I think that we could say that for an equivalence class $x \in (I + J)/I$, $x \subseteq I+J$, but in general $(I + J)/I$ is not a subset of $I+J$.