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Abstract:
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n this thesis we deal with the existence of Gröbner bases for finitely generated ide-
als in rings of polynomials over some classes of rings which are not Noetherian. The theory of
Gröbner bases is highly developed when we observe the ring of polynomials over a field or over a
Noetherian ring. The case when the base ring is non-Noetherian is less examined. In that sense,
the rings which will be of interest here are valuation rings of Krull dimension zero, valuation
domains of Krull dimension one, also the generalization of the last: Prüfer domains of Krull
dimension one. Von Neumann regular commutative rings and (p − 1)-nil-clean commutative
rings will also be a matter of discussion. The conclusions of the thesis can be applied to Bezout
and Boolean rings, as these form the subclasses of Prüfer and von Neumann regular rings,
respectively. The thesis is mostly focused on rings of polynomials with one indeterminate. |