If a K-ring S is constructed
from a K-ring R by adjoining certain new generators and relations, then
the S-bimodule ΩK(S) with a universal K-derivation d : S → ΩK(S) can
be constructed from the corresponding R-bimodule ΩK(R) by extending
scalars to S, and adjoining formal derivatives of the new generators and
relations. By studying this bimodule it is shown that a large number of natural
universal constructions preserve the class of right hereditary K-rings (K
semisimple Artinian), including the constructions of universal localization
(which had resisted earlier techniques) and certain direct limits of known
constructions. The same technique gives information on Euler characteristics of
modules (Lewin-Schreier formulas). To study universal localizations of a ring R
which may not contain a semisimple Artin ring K, a different technique is
used.
