  This habilitation (German variant of a PhD on top of a PhD) thesis presents
the quintessence of the ideas and experiences with Groebner Bases of Birgit
Reinert. She died unexpectedly without providing an abstract. As arXiv requires
an abstract, let us quote from the table of contents: History of Groebner
Bases. The two definitions of Groebner Bases. Generalizations. Groebner Bases
in Function Rings -- A Guide for Introducing Relations to Algebraic Structures.
Applications of Groebner Bases to Function Rings. Groebner Bases in Polynomial
Rings. Reduction Rings. Quotients of Reduction Rings. Sums of Reduction Rings.
Modules over Reduction Rings. Polynomial Rings over Reduction Rings. Function
Rings: Right Ideals and Right Standard Representations. Function Rings over
Fields. Function Rings over Reduction Rings. Function Rings over the Integers.
Right F-Modules. Ideals and Standard Representations. Two-sided Modules.
Applications of Groebner Bases: Natural Applications. Quotient Rings.
Elimination Theory. Polynomial Mappings. Systems of One-sided Equations in
Function Rings over the Integers.
