idempotent element and finite direct sum I have to show that if $R$ is a ring, $I_j $ is a non-zero right ideal of R for every j $\in J$ and $$R_R=\oplus_{j\in J}I_j$$ then the index set $J$ is finite.
I know I have use the property of idempotents element but I don't know how. Thank you for help!

The point is that if the ring has identity, then $1=\sum_{j\in F}i_j$ for some finite subset $F$ of $J$, then $r=\sum_{j\in F}i_jr$ for every other $r\in R$. Since representations in a direct sum are unique, it follows that $J=F$.
In this sense, having an identity is a sort of "compactness" condition for rings.
If the ring does not have identity, then this is simply false: $\bigoplus_{i\in \mathbb N}F$ is a counterexample for any field $F$.
