Determine the smallest subring of Q that contains 2/3.

To determine the smallest subring of \(\mathbb{Q}\) (the set of rational numbers) that contains \( \frac{2}{3} \), we must find the subring that includes \( \frac{2}{3} \) and is closed under addition, subtraction, and multiplication.

### Step-by-Step Explanation:

1. **Closure under Addition and Subtraction**: If the subring contains \( \frac{2}{3} \), it must also contain all integer multiples of \( \frac{2}{3} \), such as \( \frac{4}{3} \), \( \frac{6}{3} = 2 \), etc. Additionally, it must contain their negatives, like \( -\frac{2}{3} \).

2. **Closure under Multiplication**: Since the subring must be closed under multiplication, it must contain all products of \( \frac{2}{3} \) with any element of the subring. For instance, \( \left(\frac{2}{3}\right) \times \left(\frac{2}{3}\right) = \frac{4}{9} \) must be in the subring. 

3. **Smallest Subring**: The smallest subring must include all integer multiples of \( \frac{2}{3} \) and all rational numbers that can be formed by multiplying \( \frac{2}{3} \) with any rational number. This means the subring must contain all rational numbers of the form \( \frac{2k}{3m} \) where \( k \) and \( m \) are integers, and \( m \) is non-zero.

   However, the smallest subring must not only include multiples of \( \frac{2}{3} \) but also numbers like \( \frac{1}{3} \) to ensure that all these numbers are closed under multiplication.

4. **General Form**: The smallest subring that satisfies all these conditions is the set of all rational numbers with denominators dividing 3. Specifically, this subring consists of all numbers of the form \( \frac{k}{3^m} \), where \( k \) and \( m \) are integers. 

Thus, the smallest subring of \( \mathbb{Q} \) that contains \( \frac{2}{3} \) is:
\[
\mathbb{Z}\left[\frac{1}{3}\right] = \left\{ \frac{k}{3^m} \mid k \in \mathbb{Z}, m \in \mathbb{Z}_{\geq 0} \right\}
\]

This subring includes all rational numbers that can be expressed with denominators that are powers of 3.

### Do you have any questions or need further details?

### Related Questions:
1. What is a subring, and how does it differ from a ring?
2. How can we determine the smallest subring containing any given rational number?
3. Why does closure under multiplication require us to include numbers like \( \frac{1}{3} \)?
4. How do the properties of integers and rationals affect the structure of subrings in \( \mathbb{Q} \)?
5. Can this method be used to find the smallest subring containing other fractions, like \( \frac{1}{2} \)?

### Tip:
When dealing with subrings, always ensure the structure is closed under the necessary operations (addition, subtraction, multiplication) and contains the multiplicative identity (usually 1 in rational numbers).

Explore how to determine the smallest subring of the rational numbers (Q) that contains 2/3. This solution explains key algebraic concepts, such as closure properties in subrings, and demonstrates how to construct a subring that satisfies these properties. Suitable for undergraduate students studying abstract algebra.

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