Let I and J be ideals of the ring R such that I+J=R and I \cap J = {0}, show that R/I \cong J. 

To show that \(R/I \cong J\) given that \(I\) and \(J\) are ideals of the ring \(R\) with \(I + J = R\) and \(I \cap J = \{0\}\), we will construct an isomorphism between \(R/I\) and \(J\).

Here are the detailed steps:

1. **Use the fact \(I + J = R\)**:
   - Since \(I + J = R\), for any element \(r \in R\), there exist elements \(i \in I\) and \(j \in J\) such that \(r = i + j\).

2. **Define a mapping \(\phi: R \rightarrow J\)**:
   - Define \(\phi(r) = j\) where \(r = i + j\) and \(i \in I, j \in J\). This mapping is well-defined because if \(r = i + j\) and \(r = i' + j'\) for some \(i, i' \in I\) and \(j, j' \in J\), then \(i - i' = j' - j\).
   - Since \(i - i' \in I\) and \(j' - j \in J\), and \(I \cap J = \{0\}\), it follows that \(i = i'\) and \(j = j'\). Thus, \(j\) is uniquely determined by \(r\).

3. **Properties of \(\phi\)**:
   - \(\phi\) is a homomorphism: For \(r_1, r_2 \in R\), \(r_1 = i_1 + j_1\) and \(r_2 = i_2 + j_2\) with \(i_1, i_2 \in I\) and \(j_1, j_2 \in J\), we have:
     \[
     \phi(r_1 + r_2) = \phi((i_1 + j_1) + (i_2 + j_2)) = \phi((i_1 + i_2) + (j_1 + j_2)) = j_1 + j_2 = \phi(r_1) + \phi(r_2).
     \]
   - \(\phi(1_R) = 1_J\), where \(1_R\) and \(1_J\) are the multiplicative identities of \(R\) and \(J\) respectively.

4. **Kernel of \(\phi\)**:
   - The kernel of \(\phi\) is \(\ker(\phi) = \{r \in R \mid \phi(r) = 0\}\).
   - Since \(\phi(r) = j\) and \(j = 0\) implies \(r \in I\), \(\ker(\phi) = I\).

5. **First Isomorphism Theorem**:
   - By the First Isomorphism Theorem for rings, \(R / \ker(\phi) \cong \operatorname{Im}(\phi)\).
   - Since \(\ker(\phi) = I\) and \(\operatorname{Im}(\phi) = J\), we have \(R / I \cong J\).

Hence, we have shown that \(R/I \cong J\).

Would you like any further details or have any questions?

Here are 5 related questions you might find interesting:
1. What is the First Isomorphism Theorem for rings and how is it used in algebra?
2. Can you provide an example of specific ideals \(I\) and \(J\) in a ring \(R\) where \(I + J = R\) and \(I \cap J = \{0\}\)?
3. How do the concepts of quotient rings and isomorphisms help in simplifying complex algebraic structures?
4. What are the applications of the isomorphism \(R/I \cong J\) in solving polynomial equations?
5. Can you explain the significance of the condition \(I \cap J = \{0\}\) in the context of module theory?

**Tip:** When proving isomorphisms, it's crucial to ensure that the mapping is well-defined, bijective, and respects the ring operations (addition and multiplication).

Explore the proof that R/I is isomorphic to J in ring theory, given the conditions I + J = R and I ∩ J = {0}. This detailed solution utilizes the First Isomorphism Theorem for Rings and demonstrates the construction of an isomorphism between these quotient structures.

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