In mathematics, specifically commutative algebra, a divided power structure is a way of introducing items with similar properties as expressions of the form
A divided power structure (or PD-structure, after the French puissances divisées) on I is a collection of maps
when it is clear what divided power structure is meant.
Homomorphisms of divided power algebras are ring homomorphisms that respects the divided power structure on its source and target.
More generally, if M is an A-module, there is a universal A-algebra, called with PD ideal and an A-linear map (The case of divided power polynomials is the special case in which M is a free module over A of finite rank.)
If I is any ideal of a ring A, there is a universal construction which extends A with divided powers of elements of I to get a divided power envelope of I in A.
The divided power envelope is a fundamental tool in the theory of PD differential operators and crystalline cohomology, where it is used to overcome technical difficulties which arise in positive characteristic.