Could you explain how to show 9.48 in great detail? I also included a list of theorems and definitions in my textbook as a reference. Really appreciate your help! Definition 9.9. Let $ R $ be a ring. An ideal $ M $ of $ R $ is said to be **maximal** if1. $ M \neq R $; and2. If $ I $ is an ideal of $ R $ containing $ M $, then $ I = M $ or $ I = R $.Example 9.27. Let $ R = \mathbb{Z} $ and let $ n $ be a nonnegative integer. Then we claim that $ (n) $ is a maximal ideal of $ R $ if and only if $ n $ is prime. Indeed, $ (0) $ is certainly not maximal, as $ (0) \subset (2) \subset R $. Also, $ (1) $ is not maximal, since $ (1) = R $. If $ n $ is composite, say $ n = kl $, with $ 1 < k, l < n $, then we note that $ (n) \subset (k) \subset R $, so $ (n) $ is not maximal. Finally, let $ n $ be prime. Suppose that $ I $ is an ideal of $ R $ with $ (n) \subset I \subset R $. Take $ a \in I \setminus (n) $. Since $ a $ is not divisible by $ n $, and $ n $ is prime, we know that $ (a, n) = 1 $. Thus, by Corollary 2.1, we can find integers $ u $ and $ v $ such that $ au + nv = 1 $. But as $ a, n \in I $, this implies that $ 1 \in I $, hence $ I = R $, giving us a contradiction. (As we shall see shortly, there is another way to prove this.)Example 9.28. In any field, the ideal $ \{0\} $ is maximal! Remember, by Corollary 9.1, a field only has two ideals.In a commutative ring with identity, there is a nice test for maximality of ideals.Theorem 9.20. Let $ R $ be a commutative ring with identity, and $ **Question 9.48:** Find all prime ideals in each of the following rings.1. \(\mathbb{Z}_{10}$2. $\mathbb{Z}_{50}$

Find all prime ideals in each of the following rings:1 ℤ102 ℤ50

Answered: 9.48. Find all prime ideals in each of…

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9.48. Find all prime ideals in each of the following rings. 1. Z10 2. Z50

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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Confidence Intervals

When we calculate intervals in probability and statistics, it can be simply termed as finding out a confidence interval. The confidence interval is usually calculated from the data set which is observed. The value of the parameter that needs to be calculated is present here only.

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Could you explain how to show 9.48 in great detail? I also included a list of theorems and definitions in my textbook as a reference. Really appreciate your help!

Definition 9.9. Let $ R $ be a ring. An ideal $ M $ of $ R $ is said to be **maximal** if

1. $ M \neq R $; and
2. If $ I $ is an ideal of $ R $ containing $ M $, then $ I = M $ or $ I = R $.

Example 9.27. Let $ R = \mathbb{Z} $ and let $ n $ be a nonnegative integer. Then we claim that $ (n) $ is a maximal ideal of $ R $ if and only if $ n $ is prime. Indeed, $ (0) $ is certainly not maximal, as $ (0) \subset (2) \subset R $. Also, $ (1) $ is not maximal, since $ (1) = R $. If $ n $ is composite, say $ n = kl $, with $ 1 < k, l < n $, then we note that $ (n) \subset (k) \subset R $, so $ (n) $ is not maximal. Finally, let $ n $ be prime. Suppose that $ I $ is an ideal of $ R $ with $ (n) \subset I \subset R $. Take $ a \in I \setminus (n) $. Since $ a $ is not divisible by $ n $, and $ n $ is prime, we know that $ (a, n) = 1 $. Thus, by Corollary 2.1, we can find integers $ u $ and $ v $ such that $ au + nv = 1 $. But as $ a, n \in I $, this implies that $ 1 \in I $, hence $ I = R $, giving us a contradiction. (As we shall see shortly, there is another way to prove this.)

Example 9.28. In any field, the ideal $ \{0\} $ is maximal! Remember, by Corollary 9.1, a field only has two ideals.

In a commutative ring with identity, there is a nice test for maximality of ideals.

Theorem 9.20. Let $ R $ be a commutative ring with identity, and \(

**Question 9.48:** Find all prime ideals in each of the following rings.

1. $\mathbb{Z}_{10}$
2. $\mathbb{Z}_{50}$

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