Could you explain how to show 9.10 in detail? I included list of theorems and definitions from the textbook. **Definition 9.1.** Let \( R \) be a ring. Then a subring \( I \) of \( R \) is said to be an ideal if \( ir, ri \in I \) for all \( i \in I \) and \( r \in R \). We call this the absorption property.**Theorem 9.1.** Let \( R \) be a ring and \( I \) a subset of \( R \). Then \( I \) is an ideal if and only if1. \( 0 \in I \);2. \( i - j \in I \) for all \( i, j \in I \); and3. \( ir, ri \in I \) for all \( i \in I, r \in R \).**Example 9.1.** Let \( n \) be any integer. Then \( n\mathbb{Z} \) is an ideal of \( \mathbb{Z} \). Indeed, we already know that it is a subring. But also, if \( nk \in n\mathbb{Z} \), then for any integer \( r, r(nk) = n(rk) \in n\mathbb{Z} \).**Example 9.2.** Let \( I \) be the set of all polynomials \( f(x) \in \mathbb{R}[x] \) such that \( f(0) = 0 \). We claim that \( I \) is an ideal in \( \mathbb{R}[x] \). Certainly \( I \) contains the zero polynomial. Also, if \( f(0) = g(0) = 0 \), then \( (f-g)(0) = f(0) - g(0) = 0 \), hence \( f(x) - g(x) \in I \). Also, if \( f(0) = 0 \) and \( h(x) \in \mathbb{R}[x] \), then \( h(0)f(0) = h(0)0 = 0 \). Hence, \( h(x)f(x) \in I \).**Example 9.3.** Let \( I \) be the set of all polynomials in \( \mathbb{Z}[x] \) ### Problem 3.42Let \( G \) be the set of all sequences of integers \((a_1, a_2, a_3, \ldots)\).1. **Show that \( G \) is a group under** \[ (a_1, a_2, \ldots) + (b_1, b_2, \ldots) = (a_1 + b_1, a_2 + b_2, \ldots). \]2. **Let \( H \) be the set of all elements \((a_1, a_2, \ldots)\) of \( G \) such that only finitely many \( a_i \) are different from 0** (and \((0, 0, 0, \ldots) \in H\)). **Show that \( H \) is a subgroup of \( G \).**### Problem 9.9Consider the additive group \( G \) and subgroup \( H \) from Exercise 3.42. Define a multiplication operation on \( G \) via \[(a_1, a_2, \ldots)(b_1, b_2, \ldots) = (a_1b_1, a_2b_2, \ldots).\]Show that \( G \) is a ring and \( H \) is an ideal.### Problem 9.10In the preceding exercise, show that \( H \) is not a principal ideal.

This question is solved on the assumptions that the reader has proved the following1. G is a ring,…

3.42. Let G be the set of all sequences of integers (a1, a2, a3, . . .). 1. Show that G is a group under (a1, a2, . .) + (b1, b2, . .) = (a1 + b1, a2 + b2, ...). 2. Let H be the set of all elements (a1, a2, ...) of G such that only finitely many a; are different from 0 (and (0, 0, , ...) e H). Show that H is a subgroup of G. 9.9. Consider the additive group G and subgroup H from Exercise 3.42. Define a nultiplication operation on G via (a1, a2, . )(b1, b2, ...) = (a¡b1, azbɔ, ...). Show hat G is a ring and H is an ideal. 9.10. In the preceding exercise, show that H is not a principal ideal.

3.42. Let G be the set of all sequences of integers (a1, a2, a3, . . .). 1. Show that G is a group under (a1, a2, . .) + (b1, b2, . .) = (a1 + b1, a2 + b2, ...). 2. Let H be the set of all elements (a1, a2, ...) of G such that only finitely many a; are different from 0 (and (0, 0, , ...) e H). Show that H is a subgroup of G. 9.9. Consider the additive group G and subgroup H from Exercise 3.42. Define a nultiplication operation on G via (a1, a2, . )(b1, b2, ...) = (a¡b1, azbɔ, ...). Show hat G is a ring and H is an ideal. 9.10. In the preceding exercise, show that H is not a principal ideal.

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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