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1. In each case explain why R is not a ring. (a) R= {0, 1,2,3,...}, operations of Z (b) R = 27 (c) R = the set of all mappings f: R→ R; addition as in Example 4 but using composition as the multiplication

1. In each case explain why R is not a ring. (a) R= {0, 1,2,3,...}, operations of Z (b) R = 27 (c) R = the set of all mappings f: R→ R; addition as in Example 4 but using composition as the multiplication

Advanced Engineering Mathematics
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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From an abstract algebra class in the rings chapter. Just the one circled in red please

### Transcription for Educational Website **1. In each case explain why \( R \) is not a ring.** **(a)** \( R = \{0, 1, 2, 3, \cdots\} \), operations of \( \mathbb{Z} \) **(b)** \( R = 2\mathbb{Z} \) **(c)** \( R = \) the set of all mappings \( f : \mathbb{R} \rightarrow \mathbb{R} \); addition as in Example 4 but using composition as the multiplication --- #### Explanation: This exercise delves into understanding mathematical structures known as rings. The task is to explain why the given examples do not satisfy the properties required of a ring. Each example presents a different kind of mathematical set and operation. Understanding the properties that define a ring is crucial for tackling this problem. A ring requires closure under addition and multiplication, contains an additive identity, has additive inverses, and multiplication is associative, among other properties.
Transcribed Image Text:### Transcription for Educational Website **1. In each case explain why \( R \) is not a ring.** **(a)** \( R = \{0, 1, 2, 3, \cdots\} \), operations of \( \mathbb{Z} \) **(b)** \( R = 2\mathbb{Z} \) **(c)** \( R = \) the set of all mappings \( f : \mathbb{R} \rightarrow \mathbb{R} \); addition as in Example 4 but using composition as the multiplication --- #### Explanation: This exercise delves into understanding mathematical structures known as rings. The task is to explain why the given examples do not satisfy the properties required of a ring. Each example presents a different kind of mathematical set and operation. Understanding the properties that define a ring is crucial for tackling this problem. A ring requires closure under addition and multiplication, contains an additive identity, has additive inverses, and multiplication is associative, among other properties.
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