zerodivisor

Zero divisor (often written zerodivisor) is a term from ring theory. In a ring R, a nonzero element a is called a zero divisor if there exists a nonzero b in R with ab = 0. In noncommutative rings one also distinguishes left zero divisors (there exists nonzero b with ab = 0) and right zero divisors (there exists nonzero c with ca = 0). In a commutative ring these two notions coincide.

Examples help illustrate the concept. In the ring Z/6Z, the classes of 2, 3, and 4 are zero divisors because 2·3 ≡ 0, 3·2 ≡ 0, and 4·3 ≡ 0 modulo 6. In an integral domain such as Z or the polynomial ring k[x] over a field k, the only zero divisor is 0. In the product ring R × S, elements of the form (r,0) or (0,s) with the other component nonzero are zero divisors. In the matrix ring M_n(F) with n > 1, every singular matrix is a zero divisor, while only invertible matrices are not. The class of x in k[x]/(x^2) is a zero divisor since x^2 = 0.

Basic properties follow. Units are never zero divisors: if u is a unit and ua = 0, then a = 0. The nonzero divisors form a multiplicative set, and localizing R at this set yields the total quotient ring (the ring of fractions). Nilpotent elements (a^n = 0 with a ≠ 0) are typically zero divisors as well.

Zero divisors express how far a ring is from being a domain and connect to various topics in algebra and geometry.