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4. Let R be a ring. (a) Let x, y, z € R. Prove that, if x | y and y | z, then x | z. (b) Prove that u € R is a unit if and only if u | 1R. (Hint: this is all about definitions.)

4. Let R be a ring. (a) Let x, y, z € R. Prove that, if x | y and y | z, then x | z. (b) Prove that u € R is a unit if and only if u | 1R. (Hint: this is all about definitions.)

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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**Problem 4.** Let \( R \) be a ring. (a) Let \( x, y, z \in R \). Prove that if \( x \,|\, y \) and \( y \,|\, z \), then \( x \,|\, z \). (b) Prove that \( u \in R \) is a unit if and only if \( u \,|\, 1_{R} \). *Hint: this is all about definitions.* --- This exercise is designed to challenge your understanding of the definitions and properties within ring theory. In part (a), you are asked to prove a transitivity property of divisibility in a ring. In part (b), your task is to show the equivalence between a unit element and its ability to divide the multiplicative identity in the ring.
Transcribed Image Text:**Problem 4.** Let \( R \) be a ring. (a) Let \( x, y, z \in R \). Prove that if \( x \,|\, y \) and \( y \,|\, z \), then \( x \,|\, z \). (b) Prove that \( u \in R \) is a unit if and only if \( u \,|\, 1_{R} \). *Hint: this is all about definitions.* --- This exercise is designed to challenge your understanding of the definitions and properties within ring theory. In part (a), you are asked to prove a transitivity property of divisibility in a ring. In part (b), your task is to show the equivalence between a unit element and its ability to divide the multiplicative identity in the ring.
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