Arithmetic and Polynomials over Fuzzy Rings

Andreas Dress and Walter Wenzel

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Submission date: 20. Dec. 2007
published in: Communications in algebra, 43 (2015) 3, p. 1207-1231 
DOI number (of the published article): 10.1080/00927872.2013.865044
Bibtex
MSC-Numbers: 08A72, 08A05, 11C08, 13F20, 14A05
Keywords and phrases: field, homomorphism, semiring, fuzzy ring, Grassmann-Plücker map, matroid with coefficients, Zariski Topology, Algebraic and Tropical Geometry

Abstract:
We study the arithmetic in fuzzy rings and related structures as well as polynomials over these coefficient domains. Fuzzy rings are an appropriate tool for a unified approach to representable, oriented, and valuated matroids. On the other hand, the relatively new field of "Tropical Geometry" as well as "Classical Algebraic Geometry" exhibit several analogies. The methods developed in the present paper will in future research also be fundamental for a unified treatment of these disciplines.