Sum of Ideals is Ideal/Corollary
Jump to navigation
Jump to search
Corollary to Sum of Ideals is Ideal
Let $J_1$ and $J_2$ be ideals of a ring $\struct {R, +, \circ}$.
Let $J = J_1 + J_2$ be an ideal of $R$ where $J_1 + J_2$ is as defined in subset product.
Then:
- $J_1 \subseteq J_1 + J_2$
- $J_2 \subseteq J_1 + J_2$
Proof
From Sum of Ideals is Ideal we have that $j$ is an ideal of $R$.
Then:
- $0_R \in J_1 + J_2$
and so:
- $\forall x \in J_1: x + 0_R = x \in J_1 + J_2$
Similarly for $J_2$.
$\blacksquare$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 21$. Ideals: Theorem $40$