nonzerodivisor

A nonzerodivisor is an element a of a ring R with the property that multiplication by a is injective. More precisely, a is a nonzerodivisor if a ≠ 0 and for every r ∈ R, ar = 0 implies r = 0. Equivalently, the left multiplication map L_a: R → R, r ↦ ar, is injective. In a commutative ring with 1 this is the same as saying ab = 0 implies b = 0 for all b when a ≠ 0.

When R acts on an R-module M, an element a ∈ R is a nonzerodivisor on M if the map M → M, m ↦ a m, is injective. In particular, if M = R, the previous definition specializes to the ring case.

Relation to zero divisors: an element that is not a nonzerodivisor may be a zero divisor. A ring with no zero divisors is called an integral domain (in the commutative sense). In such rings every nonzero element is a nonzerodivisor.

Examples: In the ring of integers Z, every nonzero n is a nonzerodivisor. In the ring Z/6Z, the element 2 is not a nonzerodivisor since 2 · 3 ≡ 0 mod 6. In the polynomial ring k[x], the element x is a nonzerodivisor because x f(x) = 0 implies f = 0.

Notes: In commutative algebra the term regular element is often used synonymously; a sequence of nonzerodivisors that remains nonzero is called a regular sequence. The concept generalizes to modules and is central to depth, dimension, and homological properties.