14. Let Tbe the ring in Example 8. LetS=SET|f{2) = 0f. Prove that Sis a subring of T.
14. Let Tbe the ring in Example 8. LetS=SET|f{2) = 0f. Prove that Sis a subring of T.
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#14 on 3.1 on the picture I sent.
![Here is a transcription of the provided page from "Abstract Algebra: An Introduction" by Thomas W. Hungerford:
---
### 3.1 Definition and Examples of Rings
**Page 55:**
10. Is \( S = \{ (a, b) \mid a + b = 0 \} \) a subring of \( \mathbb{Z} \times \mathbb{Z} \)? Justify your answer.
11. Let \( S \) be the subset of \( M(\mathbb{R}) \) consisting of all matrices of the form \(\begin{pmatrix} a & b \\ 0 & a \end{pmatrix} \).
- (a) Prove that \( S \) is a ring.
- (b) Show that \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) is a right identity in \( S \) (meaning that \( A J = A \) for every \( A \) in \( S \)).
- (c) Show that \( J \) is not a left identity in \( S \) by finding a matrix \( B \) in \( S \) such that \( J B \neq B \).
*For more information about \( S \), see Exercise 41.*
12. Let \( \mathbb{Z}[i] \) denote the set \(\{ a + bi \mid a, b \in \mathbb{Z} \}\). Show that \( \mathbb{Z}[i] \) is a subring of \( \mathbb{C} \).
13. Let \( \mathbb{Z}[\sqrt{2}] \) denote the set \(\{ a + b\sqrt{2} \mid a, b \in \mathbb{Z} \}\). Show that \( \mathbb{Z}[\sqrt{2}] \) is a subring of \( \mathbb{R} \). [See Example 20.]
14. Let \( T \) be the ring in Example 8. Let \( S = \{ \frac{a}{b} \in \mathbb{Q} \mid a \equiv b \mod 2 \} \). Prove that \( S \) is a subring of \( T \).
15. Write out the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27260fae-539c-4ca6-9fed-6022b8026087%2F0770a08c-d101-4cad-972f-511aeee61219%2Fnbtk6f_processed.png&w=3840&q=75)
Transcribed Image Text:Here is a transcription of the provided page from "Abstract Algebra: An Introduction" by Thomas W. Hungerford:
---
### 3.1 Definition and Examples of Rings
**Page 55:**
10. Is \( S = \{ (a, b) \mid a + b = 0 \} \) a subring of \( \mathbb{Z} \times \mathbb{Z} \)? Justify your answer.
11. Let \( S \) be the subset of \( M(\mathbb{R}) \) consisting of all matrices of the form \(\begin{pmatrix} a & b \\ 0 & a \end{pmatrix} \).
- (a) Prove that \( S \) is a ring.
- (b) Show that \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) is a right identity in \( S \) (meaning that \( A J = A \) for every \( A \) in \( S \)).
- (c) Show that \( J \) is not a left identity in \( S \) by finding a matrix \( B \) in \( S \) such that \( J B \neq B \).
*For more information about \( S \), see Exercise 41.*
12. Let \( \mathbb{Z}[i] \) denote the set \(\{ a + bi \mid a, b \in \mathbb{Z} \}\). Show that \( \mathbb{Z}[i] \) is a subring of \( \mathbb{C} \).
13. Let \( \mathbb{Z}[\sqrt{2}] \) denote the set \(\{ a + b\sqrt{2} \mid a, b \in \mathbb{Z} \}\). Show that \( \mathbb{Z}[\sqrt{2}] \) is a subring of \( \mathbb{R} \). [See Example 20.]
14. Let \( T \) be the ring in Example 8. Let \( S = \{ \frac{a}{b} \in \mathbb{Q} \mid a \equiv b \mod 2 \} \). Prove that \( S \) is a subring of \( T \).
15. Write out the
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