#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

Name: #14 on 3.1 on the picture I sent. Here is a transcription of the provi
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Description: #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 subr

To show that S=f∈T|f(2)=0 is a subring of the ring T of all the continuous real functions we must…

Answered: 14. Let Tbe the ring in Example 8.…

Math

Advanced Math

14. Let Tbe the ring in Example 8. LetS=SET|f{2) = 0f. Prove that Sis a subring of T.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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