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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 \(a\in R\).

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

1 Please note that if \(R\) is commutative, every left zero divisor is also a right zero divisor.

2 Please note that this is equivalent to \(a\cdot b=0\Leftrightarrow a=0 \vee b=0\).

| | | | | created: 2014-08-31 13:20:40 | modified: 2019-06-29 08:50:48 | by: bookofproofs | references: [677]

1.Proposition: Cancellation Law

2.Definition: Zero Ring

3.Theorem: Construction of Fields from Integral Domains

4.Theorem: Finite Integral Domains are Fields

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)

[677] Modler, Florian; Kreh, Martin: “Tutorium Algebra”, Springer Spektrum, 2013