Sum of Ideals is Ideal

Theorem
Let $$J_1$$ and $$J_2$$ be ideals of a ring $$\left({R, +, \circ}\right)$$.

Then:
 * $$J = J_1 + J_2$$ is an ideal of $$R$$

where $$J_1 + J_2$$ is as defined in subset product.

General Result
Let $$J_1, J_2, \ldots, J_n$$ be ideals of a ring $$\left({R, +, \circ}\right)$$.

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.

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

Proof
By definition, $$\left({R, +}\right)$$ is an abelian group.

So from Subgroup Product of Abelian Subgroups, we have that:
 * $$\left({J, +}\right) = \left({J_1, +}\right) + \left({J_2, +}\right)$$

is itself a subgroup of $$R$$.

Now consider $$a \circ b$$ where $$a \in J, b \in R$$.

Then:

$$ $$ $$

Similarly, $$b \circ a \in J_1 + J_2$$

So by definition $$J_1 + J_2$$ is an ideal of $$R$$.

Proof of General Result
Let $$J_1, J_2, \ldots, J_n$$ be ideals of a ring $$\left({R, +, \circ}\right)$$.

Proof by induction:

For all $$n \in \N^*$$, let $$P \left({n}\right)$$ be the proposition:
 * $$J_1 + J_2 + \cdots + J_n$$ is an ideal of $$R$$.

$$P(1)$$ is true, as this just says $$J_1$$ is an ideal of $$R$$.

Basis for the Induction
$$P(2)$$ is the case:
 * $$J_1 + J_2$$ is an ideal of $$R$$

which has been proved above.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $$P \left({k}\right)$$ is true, where $$k \ge 2$$, then it logically follows that $$P \left({k+1}\right)$$ is true.

So this is our induction hypothesis:
 * $$J_1 + J_2 + \cdots + J_k$$ is an ideal of $$R$$.

Then we need to show:
 * $$J_1 + J_2 + \cdots + J_k + J_{k+1}$$ is an ideal of $$R$$.

Induction Step
This is our induction step:

Let $$J = J_1 + J_2 + \cdots + J_k$$.

From the induction hypothesis, $$J$$ is an ideal.

From the base case, $$J + J_{k+1}$$ is an ideal.

That is:
 * $$J_1 + J_2 + \cdots + J_k + J_{k+1}$$ is an ideal of $$R$$.

So $$P \left({k}\right) \implies P \left({k+1}\right)$$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $$\forall n \in \N: J_1 + J_2 + \cdots + J_n$$ is an ideal of $$R$$.

Proof of Corollary
Follows directly from the definition of join.