Decompositions of modules lacking zero sums
- 2 Downloads
A module over a semiring lacks zero sums (LZS) if it has the property that v + w = 0 implies v = 0 and w = 0. While modules over a ring never lack zero sums, this property always holds for modules over an idempotent semiring and related semirings, so arises for example in tropical mathematics.
A direct sum decomposition theory is developed for direct summands (and complements) of LZS modules: The direct complement is unique, and the decomposition is unique up to refinement. Thus, every finitely generated “strongly projective” module is a finite direct sum of summands of R (assuming the mild assumption that 1 is a finite sum of orthogonal primitive idempotents of R). This leads to an analog of the socle of “upper bound” modules. Some of the results are presented more generally for weak complements and semi-complements. We conclude by examining the obstruction to the “upper bound” property in this context.
Unable to display preview. Download preview PDF.
- M. Akian, S. Gaubert and A. Guterman, Linear independence over tropical semirings and beyond, in Tropical and Idempotent Mathematics, Contemporary Mathematics, Vol. 495, American Mathematical Society, Providence, RI, 2009, pp. 1–38.Google Scholar
- J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 36, Springer, Berlin, 1998.Google Scholar
- G. W. Brumfiel, Partially Ordered Rings and Semialgebraic Geometry, London Mathematical Society Lecture Notes, Vol. 37, Cambridge University Press, Cambridge, 1979.Google Scholar
- A. Connes and C. Consani, Characteristic 1, entropy, and the absolute point, in Noncommutative Geometry, Arithmetic, and Related Topics, Johns Hopkins University Press, Baltimore, MD, 2011, pp. 75–139.Google Scholar
- D. Dolžan and P. Oblak, Invertible and nilpotent matrices over antirings, Linear Algebra and its Applications 430 (2009), 271–278.Google Scholar
- M. Dubey, Some results on semimodules analogous to module theory, Doctoral Dissertation, University of Delhi, 2008.Google Scholar
- N. Durov, A new approach to Arakelov geometry, arXiv:0704.2030, 2007.Google Scholar
- J. Golan, Semirings and their Applications, Kluwer, Dordrecht, 1999.Google Scholar
- Z. Izhakian, Tropical arithmetic and matrix algebra, Communications in Algebra 37 (2009), 1445–1468.Google Scholar
- Z. Izhakian and L. Rowen, Supertropical algebra, Advances in Mathematics 225 (2010), 2222–2286.Google Scholar
- Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, Journal of Pure and Applied Algebra 215 (2011), 2431–2463.Google Scholar
- Z. Izhakian, M. Knebusch and L. Rowen, Supertropical linear algebra, Pacific Journal of of Mathematics 266 (2013), 43–75.Google Scholar
- Z. Izhakian, M. Knebusch and L. Rowen, Layered tropical mathematics, Journal of Algebra 416 (2014), 200–273.Google Scholar
- M. Knebusch and D. Zhang, Convexity, valuations, and Prüfer extensions in real algebra, Documenta Mathematica 10 (2005), 1–109.Google Scholar
- A. W. Macpherson, Projective modules over polyhedral semirings, arXiv:1507.07213v1, 26 Jul 2015.Google Scholar
- M. Takahashi, On the bordism categories III, Kobe University. Mathematics Seminar Notes 10 (1982), 211–236.Google Scholar
- Y. Tan, On invertible matrices over antirings, Linear Algebra and its Applications 423 (2007), 428–444.Google Scholar