Question

3.10 If U, V are ideals of R, let U + V = {u + v | u ∈ U, v ∈ V }.

Prove that U + V is also an ideal.

Answer 1

Let $u + v \in U + V$ and $r \in R$. Since $U$ is an ideal, $\{ru, ur\} \subset R$; since $V$ is an ideal, $\{rv, vr\} \subset R$. Hence, using left and right distributivity of the ring $R$,

$r(u + v) = ru + rv \in R, \\ (u + v)r = ur + vr \in R$

(the sums are elements of $R$ because the ring is stable under addition). This proves that $U + V$ is both a left and right ideal of $R$. Therefore, $U + V$ is an ideal.