Proof of Theorem 16.41. Write a complete proof of Theorem 16.41. In other words, show that if R is a ring and / is an ideal, then with coset addition and multiplication the set of cosets R/I is a ring.
Proof of Theorem 16.41. Write a complete proof of Theorem 16.41. In other words, show that if R is a ring and / is an ideal, then with coset addition and multiplication the set of cosets R/I is a ring.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Prove theorem 16.41, using the attached definition of quotient rings.
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![**Transcription for Educational Website**
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**(b)** Let \( r, s \in R \), then the following are equivalent:
1. \( r + I = s + I \),
2. \( s - r \in I \),
3. \( s = r + a \) for some \( a \in I \).
**Proof.** These results repeat Corollary 5.2 and Lemma 5.6 for the case of a subgroup of an abelian group written additively. □
---
**Definition 16.39 (Quotient rings).** Let \( (R, +, \cdot) \) be a ring, and let \( I \) be a (two-sided) ideal of \( R \). Recall that \( R/I = \{ r + I \mid r \in R \} \), and \( (R/I, +) \)—where \( (r + I) + (s + I) \) is defined to be \( (r + s + I) \)—is an abelian group. Make \( (R/I, +, \cdot) \) into a ring by defining
\[
(r + I) \cdot (s + I) = rs + I.
\]
\( (R/I, +, \cdot) \) is called the quotient ring of \( R \) by \( I \) or the factor ring of \( R \) by \( I \) or the residue class ring of \( R \) modulo \( I \).
---
**Lemma 16.40.** The multiplication of cosets defined in Definition 16.39 is well defined.
**Proof.** Since each coset has many aliases, we have to show that our definition of coset multiplication does not depend on the particular coset representative chosen. To this end, assume that \( r + I = r' + I \) and \( s + I = s' + I \). We have to show that \( rs + I = r's' + I \). From \( r + I = r' + I \) and \( s + I = s' + I \), we get that \( r' = r + x \) and \( s' = s + y \) with \( x, y \in I \). Now
\[
r's' + I = (](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5625dedc-661f-4633-8347-e309e0fcabe9%2F1d740403-1d94-428d-8322-7a56034cecb0%2Fo2fy2xt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website**
---
**(b)** Let \( r, s \in R \), then the following are equivalent:
1. \( r + I = s + I \),
2. \( s - r \in I \),
3. \( s = r + a \) for some \( a \in I \).
**Proof.** These results repeat Corollary 5.2 and Lemma 5.6 for the case of a subgroup of an abelian group written additively. □
---
**Definition 16.39 (Quotient rings).** Let \( (R, +, \cdot) \) be a ring, and let \( I \) be a (two-sided) ideal of \( R \). Recall that \( R/I = \{ r + I \mid r \in R \} \), and \( (R/I, +) \)—where \( (r + I) + (s + I) \) is defined to be \( (r + s + I) \)—is an abelian group. Make \( (R/I, +, \cdot) \) into a ring by defining
\[
(r + I) \cdot (s + I) = rs + I.
\]
\( (R/I, +, \cdot) \) is called the quotient ring of \( R \) by \( I \) or the factor ring of \( R \) by \( I \) or the residue class ring of \( R \) modulo \( I \).
---
**Lemma 16.40.** The multiplication of cosets defined in Definition 16.39 is well defined.
**Proof.** Since each coset has many aliases, we have to show that our definition of coset multiplication does not depend on the particular coset representative chosen. To this end, assume that \( r + I = r' + I \) and \( s + I = s' + I \). We have to show that \( rs + I = r's' + I \). From \( r + I = r' + I \) and \( s + I = s' + I \), we get that \( r' = r + x \) and \( s' = s + y \) with \( x, y \in I \). Now
\[
r's' + I = (
Transcribed Image Text:**Transcription for an Educational Website**
Title: Understanding Homomorphism and Ring Structures
**Proof:**
The homomorphism diagram of Figure 16.3 assists in visualizing the theorem's statement. To reinforce understanding, Problem 16.2.16 requests that readers use the proof of Theorem 11.38 as a reference to develop a complete proof.
**Example 16.46:**
Consider the homomorphism \(\phi: \mathbb{Z} \rightarrow \mathbb{Z}/12\mathbb{Z}\) defined by \(\phi(m) = m \mod 12\). Here, \(\phi\) is a ring homomorphism, and \(\ker(\phi) = 12\mathbb{Z}\). In the ring \(\mathbb{Z}/12\mathbb{Z}\), the ideal generated by 4 is \(\{0, 4, 8\}\), and \(\{0, 4, 8\} \cong \mathbb{Z}/3\mathbb{Z}\).
The homomorphism diagram further illustrates these concepts. Applying the homomorphism theorem 16.45, \(\mathbb{Z}/12\mathbb{Z} \, / \, \{0, 4, 8\} \cong \mathbb{Z}/3\mathbb{Z}\), and \(\mathbb{Z}/\mathbb{Z}12)/\{4\} \cong \mathbb{Z}/3\mathbb{Z}\).
**Problems:**
16.2.1. **Proof of Theorem 16.41:**
Compose a full proof of Theorem 16.41. This involves demonstrating that if \(R\) is a ring and \(I\) is an ideal, then the set of cosets \(R / I\) forms a ring under coset addition and multiplication.
16.2.2. **Problem:**
Let \(R = \mathbb{Z}/36\mathbb{Z}\). Determine \(R/(5)\).
**Diagram Explanation:**
Though the diagram is not visible in this transcription, Figure 16.3 presumably shows a visual model of the homomorphism process, illustrating the relationship between the rings \(\mathbb{Z}\), \(\mathbb{Z}/12\
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