Proof: Addition of Ideals is Well-Defined

In ring theory, we often define the sum of two ideals in a ring. The sum of two ideals $I$ and $J$ in a ring $R$ is defined as:

$I + J = \{ a + b \mid a \in I, b \in J \}$

We need to prove that this sum is indeed an ideal, and that it is well-defined. To be well-defined, the sum of ideals must satisfy two main conditions:

1. Closure under addition: If $x, y \in I + J$, then $x + y \in I + J$.
2. Closure under multiplication by any element in $R$: If $r \in R$ and $x \in I + J$, then $r \cdot x \in I + J$.

We will prove both properties step by step.

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Step 1: Closure under addition

Let $x, y \in I + J$. This means that:

$x = a_1 + b_1 \quad \text{and} \quad y = a_2 + b_2$ where $a_1, a_2 \in I$ and $b_1, b_2 \in J$.

Now, consider the sum $x + y$:

$x + y = (a_1 + b_1) + (a_2 + b_2) = (a_1 + a_2) + (b_1 + b_2)$

Since $I$ is an ideal, it is closed under addition, so $a_1 + a_2 \in I$. Similarly, since $J$ is an ideal, it is closed under addition, so $b_1 + b_2 \in J$.

Thus, $x + y = (a_1 + a_2) + (b_1 + b_2) \in I + J$, showing that $I + J$ is closed under addition.

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Step 2: Closure under multiplication by any element in $R$

Let $r \in R$ and $x \in I + J$. Then, $x = a + b$, where $a \in I$ and $b \in J$.

Now, consider the product $r \cdot x$:

$r \cdot x = r \cdot (a + b) = r \cdot a + r \cdot b$

Since $I$ is an ideal, $r \cdot a \in I$, and since $J$ is an ideal, $r \cdot b \in J$.

Thus, $r \cdot x = r \cdot a + r \cdot b \in I + J$, showing that $I + J$ is closed under multiplication by any element in $R$.

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Conclusion

Since both closure under addition and closure under multiplication by any element of $R$ hold, $I + J$ is an ideal of $R$. Moreover, the operation of adding ideals is well-defined, as it produces a set that satisfies the properties of an ideal.

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Next Steps

1. What are some examples of sums of ideals in specific rings (e.g., $\mathbb{Z}$ or $\mathbb{R}$)?
2. How does the sum of two prime ideals behave in a commutative ring?
3. What is the relationship between the sum of two ideals and the intersection of those ideals?
4. How does the sum of ideals relate to the ideal generated by two elements in a ring?
5. Can the sum of two maximal ideals be an ideal in some rings?

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Tip: Always verify that the sum of ideals is closed under both addition and multiplication by any element of the ring to ensure that it satisfies the properties of an ideal.