Let I be an ideal of the ring R and J be an ideal of the ring S, show that the mapping defined by f:R \times S \rightarrow (R/I) \times (S/J) with f(r,s) = (r+I,s+J) is surjective.

To show that the mapping \( f: R \times S \rightarrow (R/I) \times (S/J) \) defined by \( f(r,s) = (r+I, s+J) \) is surjective, we need to show that for every element in \((R/I) \times (S/J)\), there exists an element in \(R \times S\) that maps to it.

Let's break this down step by step:

1. **Understanding the Mapping:**
   - \(R/I\) is the quotient ring of \(R\) by the ideal \(I\). This means that elements of \(R/I\) are cosets of the form \(r + I\) where \(r \in R\).
   - Similarly, \(S/J\) is the quotient ring of \(S\) by the ideal \(J\). Elements of \(S/J\) are cosets of the form \(s + J\) where \(s \in S\).
   - The product \((R/I) \times (S/J)\) consists of pairs \((r+I, s+J)\) where \(r+I \in R/I\) and \(s+J \in S/J\).

2. **Goal:**
   - We want to show that for every pair \((a + I, b + J) \in (R/I) \times (S/J)\), there exists a pair \((r, s) \in R \times S\) such that \(f(r, s) = (a + I, b + J)\).

3. **Constructing the Preimage:**
   - Given any \((a + I, b + J) \in (R/I) \times (S/J)\), consider the pair \((a, b) \in R \times S\).
   - Applying the mapping \(f\), we have \( f(a, b) = (a + I, b + J) \).

4. **Verifying Surjectivity:**
   - For any \((a + I, b + J) \in (R/I) \times (S/J)\), we found that \(f(a, b) = (a + I, b + J)\).
   - This shows that every element \((a + I, b + J)\) in \((R/I) \times (S/J)\) has a preimage \((a, b)\) in \(R \times S\).
   - Therefore, the mapping \(f\) is surjective.

In conclusion, the mapping \( f: R \times S \rightarrow (R/I) \times (S/J) \) defined by \( f(r, s) = (r + I, s + J) \) is indeed surjective because every element in \((R/I) \times (S/J)\) has a corresponding element in \(R \times S\) that maps to it.

Would you like any further details or have any questions? 

Here are 5 related questions you might find interesting:
1. What is the kernel of the mapping \(f: R \times S \rightarrow (R/I) \times (S/J)\)?
2. How can the surjectivity of \(f\) be used to prove properties of the quotient rings \(R/I\) and \(S/J\)?
3. Can you give an example of a specific ring \(R\), an ideal \(I\), and their quotient ring \(R/I\)?
4. How does the concept of ideals in ring theory relate to the notion of normal subgroups in group theory?
5. What are some applications of quotient rings in algebraic geometry or number theory?

**Tip:** When working with quotient structures in algebra, it's often helpful to consider specific examples to build intuition before tackling more abstract proofs.

Explore the proof demonstrating the surjectivity of the mapping f: R x S → (R/I) x (S/J), where R and S are rings, I and J are ideals, and (R/I) and (S/J) are quotient rings. Understand how every element in (R/I) x (S/J) has a preimage in R x S.

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