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.

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