 If "A" is any ring, there exists a divided power ring

consisting of "divided power polynomials" in the variables

that is sums of "divided power monomials" of the form

with formula_31.Here the divided power ideal is the set of divided power polynomials with constant coefficient 0.More generally, if "M" is an "A"-module, there is a universal "A"-algebra, called

with PD ideal

and an "A"-linear map

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".