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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 divisor^{1}.
A commutative ring \(R\), which is not the zero ring and in which \(0\) is the only zero divisor^{2} is called integral domain.
Table of Contents
 Proposition: Cancellation Law
 Definition: Zero Ring
 Theorem: Construction of Fields from Integral Domains
 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
 Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013
Footnotes