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9.18. Let Ij C I2 C I3 C …… be ideals of R. Let I = U, In. 1. Show that I is an ideal of R. 2. Suppose that R/I is commutative. Show that for every a, b e R, there exists an n e N such that ab – ba e In.

9.18. Let Ij C I2 C I3 C …… be ideals of R. Let I = U, In. 1. Show that I is an ideal of R. 2. Suppose that R/I is commutative. Show that for every a, b e R, there exists an n e N such that ab – ba e In.

Advanced Engineering Mathematics
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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Could you explain how to show 9.18 in detail? I included list of theorems and definitions from the textbook.

**Definition 9.3.** Let \( R \) be a ring and \( I \) an ideal of \( R \). The **factor ring** (or quotient ring), \( R/I \), is the set of all left cosets \(\{a + I : a \in R\}\), together with the operations \((a + I) + (b + I) = a + b + I\) and \((a + I)(b + I) = ab + I\), for all \( a, b \in R \). **Theorem 9.6.** For any ring \( R \) and ideal \( I \), the factor ring \( R/I \) is a ring. **Example 9.9.** Let \( R = \mathbb{Z} \) and \( I = (5) = 5\mathbb{Z} \). Then \( R/I = \{0 + I, 1 + I, 2 + I, 3 + I, 4 + I\}\) and, for instance, \[ (2 + I) + (4 + I) = 6 + I = 1 + I \quad \text{and} \quad (3 + I)(4 + I) = 12 + I = 2 + I. \] **Example 9.10.** Let \( R = M_2(\mathbb{Z}) \) and let \( I \) be the ideal consisting of all matrices whose entries are even. Then notice that for any \( a_{ij} \in \mathbb{Z} \), we have: \[ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} + I = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} + I, \] where \( b_{ij} = 0 \) if \( a_{ij} \) is even and 1 if \( a_{ij} \) is odd. Thus, \( R/I \) consists of the sixteen different elements \[ \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}
Transcribed Image Text:**Definition 9.3.** Let \( R \) be a ring and \( I \) an ideal of \( R \). The **factor ring** (or quotient ring), \( R/I \), is the set of all left cosets \(\{a + I : a \in R\}\), together with the operations \((a + I) + (b + I) = a + b + I\) and \((a + I)(b + I) = ab + I\), for all \( a, b \in R \). **Theorem 9.6.** For any ring \( R \) and ideal \( I \), the factor ring \( R/I \) is a ring. **Example 9.9.** Let \( R = \mathbb{Z} \) and \( I = (5) = 5\mathbb{Z} \). Then \( R/I = \{0 + I, 1 + I, 2 + I, 3 + I, 4 + I\}\) and, for instance, \[ (2 + I) + (4 + I) = 6 + I = 1 + I \quad \text{and} \quad (3 + I)(4 + I) = 12 + I = 2 + I. \] **Example 9.10.** Let \( R = M_2(\mathbb{Z}) \) and let \( I \) be the ideal consisting of all matrices whose entries are even. Then notice that for any \( a_{ij} \in \mathbb{Z} \), we have: \[ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} + I = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} + I, \] where \( b_{ij} = 0 \) if \( a_{ij} \) is even and 1 if \( a_{ij} \) is odd. Thus, \( R/I \) consists of the sixteen different elements \[ \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}
**Problem 9.18**: Let \( I_1 \subseteq I_2 \subseteq I_3 \subseteq \cdots \) be ideals of \( R \). Let \( I = \bigcup_{n=1}^{\infty} I_n \). 1. Show that \( I \) is an ideal of \( R \). 2. Suppose that \( R/I \) is commutative. Show that for every \( a, b \in R \), there exists an \( n \in \mathbb{N} \) such that \( ab - ba \in I_n \).
Transcribed Image Text:**Problem 9.18**: Let \( I_1 \subseteq I_2 \subseteq I_3 \subseteq \cdots \) be ideals of \( R \). Let \( I = \bigcup_{n=1}^{\infty} I_n \). 1. Show that \( I \) is an ideal of \( R \). 2. Suppose that \( R/I \) is commutative. Show that for every \( a, b \in R \), there exists an \( n \in \mathbb{N} \) such that \( ab - ba \in I_n \).
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