B. 20. Show that the subset R= {0, 3, 6, 9, 12, 15} of Zu is a subring. Does Rhave an identity? 21. Show that the subset S = {0,2, 4, 6, 8} of Zyo isa subring. Does S'have an identity?

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#21 on the picture I sent. 

**Chapter: Rings**

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 \( S = U + E \cdot T/(1/2 \cdot 0) \). Prove that \( S \) is a subring of \( T \).

15. Write out the addition and multiplication tables for:
    (a) \( \mathbb{Z}_2 \times \mathbb{Z}_2 \)
    (b) \( \mathbb{Z}_2 \times \mathbb{Z}_3 \)
    (c) \( \mathbb{Z}_3 \times \mathbb{Z}_2 \)

16. Let \( A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \) and \( 0 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \) in \( M(R) \). Let \( S \) be the set of all matrices \( B \) such that \( AB = 0 \).
    (a) List three matrices in \( S \). [Many correct answers are possible.]
    (b) Prove that \( S \) is a subring of \( M(R) \). [Hint: If \( B \) and \( C \) are in \( S \), show that \( B + C \) and \( BC \) are in \( S \) by computing \( A(B + C) \) and \( A(BC) \).]

17. Define a new multiplication in \( \mathbb{Z} \) by the rule \( ab = 0 \) for all \( a, b \in \mathbb{Z} \). Show that with ordinary addition and this new multiplication, \( \mathbb{Z} \) is a commutative ring.

18. Define a new multiplication in \( \mathbb{Z} \) by the rule
Transcribed Image Text:**Chapter: Rings** 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 \( S = U + E \cdot T/(1/2 \cdot 0) \). Prove that \( S \) is a subring of \( T \). 15. Write out the addition and multiplication tables for: (a) \( \mathbb{Z}_2 \times \mathbb{Z}_2 \) (b) \( \mathbb{Z}_2 \times \mathbb{Z}_3 \) (c) \( \mathbb{Z}_3 \times \mathbb{Z}_2 \) 16. Let \( A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \) and \( 0 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \) in \( M(R) \). Let \( S \) be the set of all matrices \( B \) such that \( AB = 0 \). (a) List three matrices in \( S \). [Many correct answers are possible.] (b) Prove that \( S \) is a subring of \( M(R) \). [Hint: If \( B \) and \( C \) are in \( S \), show that \( B + C \) and \( BC \) are in \( S \) by computing \( A(B + C) \) and \( A(BC) \).] 17. Define a new multiplication in \( \mathbb{Z} \) by the rule \( ab = 0 \) for all \( a, b \in \mathbb{Z} \). Show that with ordinary addition and this new multiplication, \( \mathbb{Z} \) is a commutative ring. 18. Define a new multiplication in \( \mathbb{Z} \) by the rule
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