The ?rst aim of this work is to present the main results and methods of
the theory of Noetherian semigroup algebras. These general results are
then applied and illustrated in the context of certain interesting and
important concrete classes of algebras that arise in a variety of areas
and have been recently intensively studied. One of the main motivations
for this project has been the growing int- est in the class of semigroup
algebras (and their deformations) and in the application of semigroup
theoretical methods. Two factors seem to be the cause for this. First,
this ?eld covers several important classes of algebras that recently
arise in a variety of areas. Furthermore, it provides methods to
construct a variety of examples and tools to control their structure and
properties, that should be of interest to a broad audience in algebra
and its applications. Namely, this is a rich resource of constructions
not only for the noncommutative ring theorists (and not only restricted
to Noetherian rings) but also to researchers in semigroup theory and
certain aspects of group theory. Moreover, because of the role of new
classes of Noetherian algebras in the algebraic approach in
noncommutative geometry, algebras of low dimension (in terms of the
homological or the Gelfand-Kirillov - mension) recently gained a lot of
attention. Via the applications to the Yang-Baxter equation, the
interest also widens into other ?elds, most - tably into mathematical
physics.
