Prove $I \subseteq I+J$ and $J \subseteq I+J$

Question

Let $R$ be a ring and $I$ and $J$ be the ideals of $R$. Prove Prove $I \subseteq I+J$ and $J \subseteq I+J$

I know this is very trivial, but I still need to check what I am doing is correct or not...

Let, $a \in I$. Since, $I$ and $J$ are the ideals of R, $a = a+0_R \in I + J$. Hence, $I \subseteq I + J$.

Similarly, let $b \in J$. Since, $I$ and $J$ are the ideals of R, $b = 0_R+b \in I + J$. Hence, $J \subseteq I + J$. Am I missing anything here?

Answer 1

The answer is in some sense correct but what you wrote does not really highlight the reason why what you claim is true.

"Since, $I$ and $J$ are the ideals of $R$, $a = a+0_R \in I + J$."

Okay. But what specific properties do you use here?

Note, that in fact you do not use at all that $I$ is an ideal here, and from $J$ being an ideal you merely use that $0 \in J$.

I would write instead: "Since $J$ is an ideal, we have $0_R \in J$ and thus $a = a+0_R \in I + J$."