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9.16. Let R be a ring and I a proper ideal. 1. If R is an integral domain, does it follow that R/I is an integral domain? Prove that it does, or find a counterexample. 2. If R/I is an integral domain, does it follow that R is an integral domain? Prove that it does, or find a counterexample.

9.16. Let R be a ring and I a proper ideal. 1. If R is an integral domain, does it follow that R/I is an integral domain? Prove that it does, or find a counterexample. 2. If R/I is an integral domain, does it follow that R is an integral domain? Prove that it does, or find a counterexample.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Could you explain how to show 9.16 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 \( R \). Then 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, \((2 + I) + (4 + I) = 6 + I = 1 + I\) and \((3 + I)(4 + I) = 12 + I = 2 + I\). **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} \\ \
Transcribed Image Text:**Definition 9.3:** Let \( R \) be a ring and \( I \) an ideal of \( R \). Then 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, \((2 + I) + (4 + I) = 6 + I = 1 + I\) and \((3 + I)(4 + I) = 12 + I = 2 + I\). **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} \\ \
**9.16.** Let \( R \) be a ring and \( I \) a proper ideal. 1. If \( R \) is an integral domain, does it follow that \( R/I \) is an integral domain? Prove that it does, or find a counterexample. 2. If \( R/I \) is an integral domain, does it follow that \( R \) is an integral domain? Prove that it does, or find a counterexample.
Transcribed Image Text:**9.16.** Let \( R \) be a ring and \( I \) a proper ideal. 1. If \( R \) is an integral domain, does it follow that \( R/I \) is an integral domain? Prove that it does, or find a counterexample. 2. If \( R/I \) is an integral domain, does it follow that \( R \) is an integral domain? Prove that it does, or find a counterexample.
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