Let $R$ be a commutative ring. Prove that the following holds: If $I$ and $J$ are ideals of $R$ then $I$ + $J$ is an ideal of $R$.

Question

I understand that I am supposed to prove that $I$ and $J$ are subsets of $R$ but I am unsure how to go about doing this. Any help is appreciated. Thanks

Answer 1

$I + J = \{ i + j, \; i \in I, \; j \in J \}; \tag 1$

then if

$x + y, \; x' + y' \in I + J, \; x, x' \in I, y, y' \in J, \tag 2$

$(x + y) - (x' + y') = (x - x') + (y - y') \in I + J; \tag 3$

also, if in addition

$r \in R, \tag 4$

$r(x + y) = rx + ry \in I + J, \tag 5$

since

$rx \in I, ry \in J, \tag 6$

by virtue of the hypothesis that $I$ and $J$ are ideals of $R$

(3) and (5) show that $I + J$ meets the defining requirements of an ideal in $R$.