Sum of Ideals is Ideal/General Result/Corollary

Corollary to Sum of Ideals is Ideal/General Result
Let $J_1, J_2, \ldots, J_n$ be ideals of a ring $\left({R, +, \circ}\right)$.

Let $J = J_1 + J_2 + \cdots + J_n$ be an ideal of $R$ where $J_1 + J_2 + \cdots + J_n$ is as defined in subset product.

$J$ is contained in every subring of $R$ containing $\displaystyle \bigcup_{k \mathop = 1}^n {J_k}$.

Proof
From Sum of Ideals is Ideal/General Result we have that $J$ is an ideal of $R$.

The result follows directly from the definition of join of subgroups.