Let a, b, and c be elements of a commutative ring R with a and b ‡ Ŕ. If a divides b and b divides c, the prove that a divides c. [This s called the transitive property for division in a ring R.]
Let a, b, and c be elements of a commutative ring R with a and b ‡ Ŕ. If a divides b and b divides c, the prove that a divides c. [This s called the transitive property for division in a ring R.]
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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![### Definition in Ring Theory
**Definition:**
In a commutative ring \( R \) where \( s \) and \( t \) are elements in \( R \), we say that if \( s \neq 0_R \) and \( s \) divides \( t \), then there is an element \( b \in R \) such that \( t = s \cdot b \).
### Problem Statement
1. Let \( a, b, \) and \( c \) be elements of a commutative ring \( R \) with \( a \) and \( b \neq 0_R \). If \( a \) divides \( b \) and \( b \) divides \( c \), then prove that \( a \) divides \( c \). [This is called the transitive property for division in a ring \( R \).]
### Proof
(Proof details would be provided here once drawn up.)
In this problem, we are working within the constraints of a commutative ring \( R \), dealing with the transitive property of division, an important concept in abstract algebra. The goal is to establish that the division relation is transitive among the given elements.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc158a850-76a9-4504-97b9-8593e0926539%2Fab54e2bd-bbc1-46b3-87b9-cffaea339fed%2F1o7xed_processed.png&w=3840&q=75)
Transcribed Image Text:### Definition in Ring Theory
**Definition:**
In a commutative ring \( R \) where \( s \) and \( t \) are elements in \( R \), we say that if \( s \neq 0_R \) and \( s \) divides \( t \), then there is an element \( b \in R \) such that \( t = s \cdot b \).
### Problem Statement
1. Let \( a, b, \) and \( c \) be elements of a commutative ring \( R \) with \( a \) and \( b \neq 0_R \). If \( a \) divides \( b \) and \( b \) divides \( c \), then prove that \( a \) divides \( c \). [This is called the transitive property for division in a ring \( R \).]
### Proof
(Proof details would be provided here once drawn up.)
In this problem, we are working within the constraints of a commutative ring \( R \), dealing with the transitive property of division, an important concept in abstract algebra. The goal is to establish that the division relation is transitive among the given elements.
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