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The already broad range of applications of ring theory has been
enhanced in the eighties by the increasing interest in algebraic
structures of considerable complexity, the so-called class of
quantum groups. One of the fundamental properties of quantum groups
is that they are modelled by associative coordinate rings
possessing a canonical basis, which allows for the use of
algorithmic structures based on Groebner bases to study them. This
book develops these methods in a self-contained way, concentrating
on an in-depth study of the notion of a vast class of
non-commutative rings (encompassing most quantum groups), the
so-called Poincar -Birkhoff-Witt rings. We include algorithms which
treat essential aspects like ideals and (bi)modules, the
calculation of homological dimension and of the Gelfand-Kirillov
dimension, the Hilbert-Samuel polynomial, primality tests for prime
ideals, etc.
The already broad range of applications of ring theory has been
enhanced in the eighties by the increasing interest in algebraic
structures of considerable complexity, the so-called class of
quantum groups. One of the fundamental properties of quantum groups
is that they are modelled by associative coordinate rings
possessing a canonical basis, which allows for the use of
algorithmic structures based on Groebner bases to study them. This
book develops these methods in a self-contained way, concentrating
on an in-depth study of the notion of a vast class of
non-commutative rings (encompassing most quantum groups), the
so-called Poincar -Birkhoff-Witt rings. We include algorithms which
treat essential aspects like ideals and (bi)modules, the
calculation of homological dimension and of the Gelfand-Kirillov
dimension, the Hilbert-Samuel polynomial, primality tests for prime
ideals, etc.
Integrates fundamental techniques from algebraic geometry,
localization theory and ring theory, and demonstrates how each
topic is enhanced by interaction with others, providing new results
within a common framework. Technical conclusions are presented and
illustrated with concrete examples.
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