Sum of Ideals is Ideal
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Theorem
Let $J_1$ and $J_2$ be ideals of a ring $\struct{R, +, \circ}$.
Then:
- $J = J_1 + J_2$ is an ideal of $R$
where $J_1 + J_2$ is as defined in subset product with respect to $\struct{R, +}$.
General Result
Let $J_1, J_2, \ldots, J_n$ be ideals of a ring $\struct {R, +, \circ}$.
Then:
- $J = J_1 + J_2 + \cdots + J_n$ is an ideal of $R$.
where $J_1 + J_2 + \cdots + J_n$ is as defined in subset product with respect to $\struct{R, +}$.
Corollary
- $J_1 \subseteq J_1 + J_2$
- $J_2 \subseteq J_1 + J_2$
Proof
By definition, $\struct {R, +}$ is an abelian group.
So from Subset Product of Abelian Subgroups, we have that:
- $\struct{J, +} = \struct{J_1, +} + \struct{J_2, +}$
is itself a subgroup of $R$.
Now consider $a \circ b$ where $a \in J, b \in R$.
Then:
\(\ds a \circ b\) | \(=\) | \(\ds \paren {a_1 + a_2} \circ b\) | by definition of $a$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a_1 \circ b} + \paren {a_2 \circ b}\) | as $\circ$ is distributive over $+$ | |||||||||||
\(\ds \) | \(\in\) | \(\ds J_1 + J_2\) | as $\paren {a_i \circ b} \in J_i$ because $J_i$ is an ideal of $R$ |
Similarly, $b \circ a \in J_1 + J_2$
So by definition $J_1 + J_2$ is an ideal of $R$.
$\blacksquare$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 21$. Ideals: Theorem $40$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields