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
Related questions
Question
100%
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 if
1. \( 0 \in I \);
2. \( i - j \in I \) for all \( i, j \in I \); and
3. \( ir, ri \in I \) 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 \( \mathbb{Z}[x] \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2F23e40629-1892-459c-b996-c6d9cd1fc193%2F7yv9nbi_processed.png&w=3840&q=75)
Transcribed Image Text:**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 if
1. \( 0 \in I \);
2. \( i - j \in I \) for all \( i, j \in I \); and
3. \( ir, ri \in I \) 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 3.42
Let \( G \) 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 9.9
Consider 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 9.10
In the preceding exercise, show that \( H \) is not a principal ideal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2F23e40629-1892-459c-b996-c6d9cd1fc193%2F8pqvfn_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 3.42
Let \( G \) 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 9.9
Consider 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 9.10
In the preceding exercise, show that \( H \) is not a principal ideal.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
