Quadratic structures associated to (multi)rings

Document Type : Research Paper


1 Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Brazil.

2 Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Brazil


We consider certain pairs (A, T) where A is a (multi)ring and
T ⊆ A is a multiplicative set that generates, by a convenient quotient construction,
a (multi)structure that supports a quadratic form theory: with
some natural hypotheses we generalize constructions previously presented
in [3] and [6]. This also provides some steps towards an abstract formally
real quadratic form theory (non necessarily reduced) were the forms have
general coefficients (non only units).


Main Subjects

[1] Ad´amek, J. and Rosicky, J., “Locally Presentable and Accessible Categories”, London Math. Soc. Lecture Note Ser. 189, Cambridge University Press, 1994.
[2] Lima, A., Les Groupes Speciaux. Aspects Algebriques et Combinatoires de la Theorie des Espaces d’Ordres Abstraits, Ph.D. Thesis, University Paris 7, 1996.
[3] Dickmann, M. and Miraglia, F., “Special Groups: Boolean-theoretic Methods in the Theory of Quadratic Forms”, Mem. Amer. Math. Soc. 145(689), American Mathematical Society, 2000.
[4] Dickmann, M. and Miraglia, F., Quadratic form theory over preordered von neumann-regular rings, J. Algebra 319(4) (2008), 1696-1732.
[5] Maximo Dickmann and Francisco Miraglia. “Faithfully Quadratic Rings”, Mem. Amer. Math. Soc. 238(1128), American Mathematical Society, 2015.
[6] Dickmann, M. and Petrovich, A., Real semigroups and abstract real spectra, I., Algebraic and arithmetic theory of quadratic forms, Contemp. Math. 344 (2004), 99-120.
[7] Engelking, R., “General Topology”, Sigma Series in Pure Mathematics, Heldermann Verlag, 1989.
[8] Gladki, P. and Marshall, M., Orderings and signatures of higher level on multirings and hyperfields, J. K-Theory: K-Theory Appl. Algebra Geom. Topol. 10(3) (2012), 489-518.
[9] Gladki, P. and Worytkiewicz, K., Witt rings of quadratically presentable fields, Categ. General Alg. Struct. Appl. 12(1) (2020), 1-23.
[10] Jun, J., Algebraic geometry over hyperrings, Adv. Math. 323 (2018), 142-192.
[11] Marshall, M., Real reduced multirings and multifields, J. Pure Appl. Algebra 205(2) (2006), 452-468.
[12] Marshall, M., “Spaces of Orderings and Abstract Real Spectra”, Springer, 1996.
[13] Ribeiro, H.R.d.O. and Mariano, H.L., von Neumann regular hyperrings and applications to real reduced multirings, arXiv:2101.06527, 2021.
[14] Ribeiro, H.R.d.O., Roberto, K.M.d.A., and Mariano, H.L., Functorial relationship between multirings and the various abstract theories of quadratic forms, S˜ao Paulo J. Math. Sci., 2020, https://doi.org/10.1007/s40863-020-00185-1.
[15] Viro, O., Hyperfields for tropical geometry I., hyperfields and dequantization, arXiv:1006.3034, 2010.