Zeroderivation

Zeroderivation is a term that appears in some discussions of ring theory to describe a derivation with a property related to zero divisors. It is not a universally standardized notion, and different authors may use the label with slightly different emphasis. The most common use is to refer to a derivation that vanishes on every zero divisor.

Definition: Let R be a commutative ring with identity and let d: R → R be a derivation, meaning d(a + b) = d(a) + d(b) and d(ab) = d(a)b + a d(b) for all a, b in R. The map d is called a zeroderivation if d(z) = 0 for every zero divisor z of R. If R has no nonzero zero divisors (i.e., R is a domain), the zeroderivation condition is vacuous, and every derivation is a zeroderivation.

Basic properties: If Z(R) denotes the set of zero divisors, then a zeroderivation satisfies d(Z(R)) = {0}. Consequently, the image d(R) lies in a substructure that annihilates Z(R). In particular, if the ideal generated by the zero divisors is I, then d(I) = 0. The existence of nontrivial zeroderivations depends on the ring’s zero-divisor structure; some rings admit only the zero derivation as a zeroderivation.

Examples and limitations: In a product ring R = A × B, zero divisors have a zero component. A derivation on R is typically given by a pair (d_A, d_B). To kill all zero divisors, one must have d_A = 0 on A and d_B = 0 on B, yielding the zero derivation. In many rings with nontrivial idempotents, nonzero zeroderivations are scarce or non-existent.

Context: Zeroderivations relate to the study of derivations on rings with zero divisors and to modules of differentials. They offer a lens on how derivations interact with the zero-divisor structure.