This book is based on a lecture course that I gave at the University of
Regensburg. The purpose of these lectures was to explain the role of
Kähler differential forms in ring theory, to prepare the road for their
application in algebraic geometry, and to lead up to some research
problems. The text discusses almost exclusively local questions and is
therefore written in the language of commutative alge- bra. The
translation into the language of algebraic geometry is easy for the
reader who is familiar with sheaf theory and the theory of schemes. The
principal goals of the monograph are: To display the information
contained in the algebra of Kähler differential forms (de Rham algebra)
of a commutative algebra, to int- duce and discuss "differential
invariants" of algebras, and to prove theorems about algebras with
"differential methods". The most important object we study is the module
of Kähler differentials n /R of an algebra SIR. Like the differentials
of analysis, differential modules "linearize" problems, i.e. reduce
questions about algebras (non-linear problems) to questions of linear
algebra. We are mainly interested in algebras of finite type.