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}.

- \(a\cdot b=0\), then we call \(a\) a

A commutative ring \(R\), which is not the zero ring and in which \(0\) is the only zero divisor^{2} 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]

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