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Definition: Zero Divisor and Integral Domain

Let \((R, + ,\cdot)\) be a ring with \(1\) as the identity of \((R,\cdot)\) and \(0\) as the identity of \((R,+)\). Further, let \(a\in R\).

• If \(a\neq 0\) and there exists \(b\in R, b\neq 0\) with
  • \(a\cdot b=0\), then we call \(a\) a left zero divisor in \(R\).
  • \(b\cdot a=0\), then we call \(a\) a right zero divisor in \(R\).
  • we call \(a\) a zero divisor in \(R\), if \(a\) is both, a left and a right zero divisor¹.

A commutative ring \(R\), which is not the zero ring and in which \(0\) is the only zero divisor² is called integral domain.

Table of Contents

1. Proposition: Cancellation Law
2. Definition: Zero Ring
3. Theorem: Construction of Fields from Integral Domains
4. Theorem: Finite Integral Domains are Fields

Mentioned in:

Branches: 1
Chapters: 2
Definitions: 3 4 5 6 7 8 9 10 11 12 13 14
Lemmas: 15 16 17 18
Parts: 19
Proofs: 20 21 22 23 24 25 26 27 28 29 30 31 32 33
Propositions: 34 35 36 37 38 39 40 41 42
Theorems: 43 44

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References

Bibliography

1. Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013

Footnotes