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Find all units, zero-divisors, and nilpotent elements in the following rings: a) ZOZ;

Find all units, zero-divisors, and nilpotent elements in the following rings: a) ZOZ;

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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**Task: Find all units, zero-divisors, and nilpotent elements in the following rings:** a) \( \mathbb{Z} \oplus \mathbb{Z} \); **Explanation:** This problem asks to identify different types of elements in the direct sum of the integers with itself, denoted \( \mathbb{Z} \oplus \mathbb{Z} \). In ring theory: - **Units:** Elements that have a multiplicative inverse. - **Zero-divisors:** Non-zero elements \( a \) and \( b \) such that \( ab = 0 \). - **Nilpotent elements:** Elements \( a \) such that \( a^n = 0 \) for some positive integer \( n \). In the ring \( \mathbb{Z} \oplus \mathbb{Z} \), these properties are analyzed based on the structure of the direct sum of two integer sets.
Transcribed Image Text:**Task: Find all units, zero-divisors, and nilpotent elements in the following rings:** a) \( \mathbb{Z} \oplus \mathbb{Z} \); **Explanation:** This problem asks to identify different types of elements in the direct sum of the integers with itself, denoted \( \mathbb{Z} \oplus \mathbb{Z} \). In ring theory: - **Units:** Elements that have a multiplicative inverse. - **Zero-divisors:** Non-zero elements \( a \) and \( b \) such that \( ab = 0 \). - **Nilpotent elements:** Elements \( a \) such that \( a^n = 0 \) for some positive integer \( n \). In the ring \( \mathbb{Z} \oplus \mathbb{Z} \), these properties are analyzed based on the structure of the direct sum of two integer sets.
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Step 1: Define units ,zero divisors and nilpotent element in a ring

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