Please if able explain the following definition in more detail, I understand the definition of a field in groups but don't really get a field in rings, for example, why does the definition mention 1 is inequal to 0? Definition. A ring R with identity 1, where 1 # 0, is called a division ring (or skewfield) if every nonzero element a € R has a multiplicative inverse, i.e., there existsb = R such that ab = ba = 1. A commutative division ring is called a field.

A ring with identity where , is called a division ring (or skew field) if every nonzero element…

Definition. A ring R with identity 1, where 1 # 0, is called a division ring (or skew field) if every nonzero element a € R has a multiplicative inverse, i.e., there exists be R such that ab = ba = 1. A commutative division ring is called a field.

Definition. A ring R with identity 1, where 1 # 0, is called a division ring (or skew field) if every nonzero element a € R has a multiplicative inverse, i.e., there exists be R such that ab = ba = 1. A commutative division ring is called a field.

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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Question

Please if able explain the following definition in more detail, I understand the definition of a field in groups but don't really get a field in rings, for example, why does the definition mention 1 is inequal to 0?

Definition. A ring R with identity 1, where 1 # 0, is called a division ring (or skew
field) if every nonzero element a € R has a multiplicative inverse, i.e., there exists
b = R such that ab = ba = 1. A commutative division ring is called a field.

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