Let \(R\) be a commutative ring. An element \(r\in R\) is a zero divisor if there exists a nonzero \(x\in R\) such that \(rx=0\).
Equivalently, the \(R\)-linear map \(R\to R\), \(x\mapsto rx\), is not injective, since its kernel is \(\theset{x\in R \st rx=0}\).
What is a zero divisor?