Definition
For commutative ring with , is called a zero-divisor if there exists given by with additive, , and not injective if and only if , .
Remark
If is finite, then every is either a zero-divisor, or a unit.
For commutative ring (R,+,⋅) with 1R, r∈R is called a zero-divisor if there exists φ:R→R given by φ(x)=rx with φ additive, kerφ={x∈R∣rx=0}, and φ not injective if and only if ∃x=0R, rx=0.
If R is finite, then every r∈R is either a zero-divisor, or a unit.