Zariskian Filtrations

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This text presents a theory of filtrations on associative rings, combining techniques stemming from number theory related to valuations, with facts originating in the study of rings of differential operators on varieties. It deals with the homological algebra part of the theory via an innovative use of graded ring theory applied to the Rees ring of a filtration. This leads to an approach to extensions of valuations, regularity conditions on noncommutative algebras, and geometric aspects of rings of differential operators, and provides new applications related to deformations of algebras, gauge algebras and other physics-related objects. This volume should be of interest to graduate students and researchers in different fields of mathematics and mathematical physics.