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 $\struct {R, +, \circ}$.

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 $\ds \bigcup_{k \mathop = 1}^n {J_k}$.

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.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old