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I even got a hint: consider $(I + J)/J$ in $R/J$, but I still don't know how to prove it.

edit: well, I can at least prove that $I + J$ is left ideal.

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Hint for your hint:

The idea is to show that $(I+J)/J$ is a nil left ideal of $R/J$. This means that for any $i\in I$, $j\in J$, there is a power of $i+j$ in $J$.

You can see the last step after that, right?

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    if $R/J$ what? If it is nil? But isn't it even more complicated to show that about $R/J$? – 2017-02-12
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    @PanMiroslav Sorry, mobile device typo. It should be **of** not *if*. And no, I don't think it is complicated at all. It seems like the obvious thing to do with the hint you are given. – 2017-02-12
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    if I understand, the most important part is to show that for any element $x$ in $(I + J)/J$ there is some $n$ such that $x^n = 0$ and it should be obvious. All I can think of is this: for any $n \in \mathbb{N}:(x + J)^n = (i + j + J)^n$ (because $x \in I + J$) $=(i + J)^n = i^n + J^n$ (from definition of that multiplication), but since $i \in I$ and $I, J$ are nil, we have some $k, l$ such that $i^k = 0$ and $J^l = 0$, so $(i + J)^{kl} =$ $i^k^l + J^l^k = 0^l + 0^k = 0 + 0 = 0$, right? – 2017-02-12
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    and in your answer, should it be "$i + j$ in $I + J$" in the end? – 2017-02-12
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    @PanMiroslav To answer your last comment, no, I do not see anything wrong now. I don't see where your correction would go. – 2017-02-12
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    @PanMiroslav To answer your longer comment: no, you are making a few mistakes. The expression $J^n$ is totally irrelevant to your computations. You are right that you need $x^n=0$, but $=0$ in $R/J$ means $x^n\in J$. It's not clear from what you wrote that you understand this. – 2017-02-12
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    @PanMiroslav Here's a sample computation to try to fix your misconceptions: $(i+j +J)^2=i^2+ij+ji+j^2+J$. If it happened that $i^2=0$, then we would be done since then $i^2+ij+ji+j^2+J=ij+ji+j^2+J=0+J$. – 2017-02-12
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    oh, okay, I understand this now, I thought that if $j \in J$ then $j + J = J$. And about that "It's not clear from what you wrote that you understand this" - yes, I forgot about that. Thank you, I think I got it now – 2017-02-12
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    @PanMiroslav I encourage writing $0+J$ instead of just $J$ when doing arithmetic with cosets. they are slightly different, although people sometimes use them interchangeably. – 2017-02-13