Classical valuation theory has applications in number theory and class field theory as well as
in algebraic geometry e.g. in a divisor theory for curves. But the noncommutative equivalent
is mainly applied to finite dimensional skewfields. Recently however new types of algebras
have become popular in modern algebra Weyl algebras deformed and quantized algebras quantum
groups and Hopf algebras etc. The advantage of valuation theory in the commutative case is
that it allows effective calculations bringing the arithmetical properties of the ground field
into the picture. This arithmetical nature is also present in the theory of maximal orders in
central simple algebras. Firstly we aim at uniting maximal orders valuation rings Dubrovin
valuations etc. in a common theory the theory of primes of algebras. Secondly we establish
possible applications of the noncommutative arithmetics to interesting classes of algebras
including the extension of central valuations to nice classes of quantized algebras the
development of a theory of Hopf valuations on Hopf algebras and quantum groups noncommutative
valuations on the Weyl field and interesting rings of invariants and valuations of Gauss
extensions.
