Let \(R\) be a ring. An element \(r\in R\) is a zero divisor if there exists \(a\in R\setminus\theset 0\) such that \(ar = ra = 0\).
If \(R\) is commutative, then \(r\in R\) is a zero divisor if and only if the \(R\)-linear map \[ R \to R, \qquad x \mapsto rx \] is not injective.