Determine the subring of \mathbb{Z} \times \mathbb{Z} generated by the element (1,-1)

Let's determine the subring of \( \mathbb{Z} \times \mathbb{Z} \) generated by the element \( (1, -1) \).

### Step 1: Understanding the problem
We are asked to find the subring of \( \mathbb{Z} \times \mathbb{Z} \) generated by the single element \( (1, -1) \). A subring of a ring is a subset that is closed under addition, subtraction, and multiplication, and that contains the multiplicative identity if it's required by the definition.

### Step 2: Generate elements using \( (1, -1) \)
Consider the element \( (1, -1) \) in \( \mathbb{Z} \times \mathbb{Z} \). We can generate other elements of the subring by taking integer linear combinations of \( (1, -1) \) with itself. For any integer \( n \), the element \( n(1, -1) \) is given by:
\[
n(1, -1) = (n \cdot 1, n \cdot (-1)) = (n, -n).
\]
Thus, every element in the subring generated by \( (1, -1) \) has the form \( (n, -n) \) where \( n \in \mathbb{Z} \).

### Step 3: Characterize the subring
The set of all such elements is:
\[
\{ (n, -n) \mid n \in \mathbb{Z} \}.
\]
Let's denote this set by \( S \). We need to check that \( S \) is indeed a subring of \( \mathbb{Z} \times \mathbb{Z} \):

- **Additive Closure:** For any \( (a, -a) \) and \( (b, -b) \) in \( S \),
  \[
  (a, -a) + (b, -b) = (a+b, -(a+b)),
  \]
  which is still of the form \( (c, -c) \) for some \( c \in \mathbb{Z} \).

- **Multiplicative Closure:** For any \( (a, -a) \) and \( (b, -b) \) in \( S \),
  \[
  (a, -a) \cdot (b, -b) = (ab, -ab),
  \]
  which is also of the form \( (c, -c) \) for some \( c \in \mathbb{Z} \).

- **Contains the identity element (if required):** The element \( (1, -1) \) is already in \( S \), and \( (0,0) \) is also in \( S \), as it corresponds to \( n = 0 \).

- **Subtraction:** For any \( (a, -a) \) in \( S \),
  \[
  -(a, -a) = (-a, a),
  \]
  which is also of the form \( (c, -c) \) with \( c = -a \).

### Step 4: Conclusion
The subring of \( \mathbb{Z} \times \mathbb{Z} \) generated by the element \( (1, -1) \) is the set:
\[
\{ (n, -n) \mid n \in \mathbb{Z} \}.
\]
This subring consists of all pairs \( (n, -n) \), where \( n \) is any integer. This is often denoted by \( \{(n, -n) \mid n \in \mathbb{Z}\} \) or sometimes as \( \{(n, -n) : n \in \mathbb{Z}\} \).

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Do you want any details or have any questions?

Here are some related questions you might consider:

1. What is a subring, and how is it different from a subgroup?
2. How do you find the subring generated by multiple elements in \( \mathbb{Z} \times \mathbb{Z} \)?
3. What if we generate a subring using the element \( (1, 1) \) instead?
4. How would the subring change if we consider \( \mathbb{Z} \times \mathbb{Z} \times \mathbb{Z} \)?
5. What is the structure of the subring generated by \( (2, -3) \) in \( \mathbb{Z} \times \mathbb{Z} \)?
6. Can you describe the quotient ring \( (\mathbb{Z} \times \mathbb{Z}) / S \) where \( S = \{(n, -n) \mid n \in \mathbb{Z}\} \)?
7. How would the problem change if we replaced \( \mathbb{Z} \) with another ring, like \( \mathbb{Z}_n \)?
8. What are some examples of subrings in \( \mathbb{Z} \times \mathbb{Z} \) that are not finitely generated?

**Tip:** When working with subrings, always check that your set is closed under the operations defined by the ring and contains the necessary elements like the identity and inverses.

Explore the subring generated by the element (1, -1) in the direct product ring Z x Z. Learn how to identify and characterize this subring, including its closure properties under addition, subtraction, and multiplication. Suitable for advanced undergraduate students and those interested in algebraic structures.

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