On the Jacobson Semisimple Semirings
Download PDF
Authors: A. K. BHUNIYA AND P. SARKAR
DOI: 10.46793/KgJMat2607.1105B
Abstract:
Based on the minimal and simple representations, we introduce two types of Jacobson semisimplicity, m-semisimplicity and s-semisimplicity, of a semiring S. Every m(s)-semisimple semiring is a subdirect product of m(s)-primitive semirings. It is shown that a commutative s-primitive semiring is either a two element Boolean algebra or a field. Every s-primitive semiring is isomorphic to a 1-fold transitive subsemiring of the semiring of all endomorphisms of a semimodule over a division semiring.
Keywords:
Semiring, Jacobson semisimple, faithful, primitive, transitive semiring
References:
[1] M. Akian, R. Bapat and S. Gaubert, Max-plus algebra, in: L. Hogben, R. Brualdi, A. Greenbaum, R. Mathias (Eds.), Handbook of Linear Algebra, Chapman & Hall, London, 2006.
[2] R. El Basir, J Hurt, A. Jančařik and T. Kepka, Simple commutative semirings, J. Algebra 236 (2001), 277–306. https://doi.org/10.1006/jabr.2000.8483
[3] A. Bertram and R. Easton, The tropical Nullstellensatz for congruences, Adv. Math. 308 (2017), 36–82. https://doi.org/10.1016/j.aim.2016.12.004
[4] A. K. Bhuniya and T. Mondal, Distributive lattice decompositions of semirings with a semilattice additive reduct, Semigroup Forum 80 (2010), 293–301. https://doi.org/10.1007/s00233-009-9205-6
[5] A. K. Bhuniya and T. Mondal, On the least distributive lattice congruence on a semiring with a semilattice additive reduct, Acta Math. Hungar. 147 (2015), 189–204. https://doi.org/10.1007/s10474-015-0526-5
[6] A. K. Bhuniya and P. Sarkar, On the Jacobson radical of a semiring, J. Algebra Appl. https://doi.org/10.1142/S0219498825503694
[7] D. Castella, Eléments d’algèbre linéaire tropicale, Linear Algebra Appl. 432 (2010), 1460–1474. https://doi.org/10.1016/j.laa.2009.11.005
[8] C. Chen et al., Extreme representations of semirings, Serdica Math. J. 44(3/4) (2018), 365–412. https://doi.org/10.48550/arXiv.1806.06501
[9] J. H. Conway, Regular Algebra and Finite Machines, Chapman & Hall, London, 1971.
[10] A. Connes and C. Consani, Schemes over F1 and zeta functions, Compos. Math. 146(6) (2010), 1383–1415. https://doi.org/10.1112/S0010437X09004692
[11] N. Damljanović, M. Ćirić and J. Ignjatović, Bisimulations for weighted automata over an additively idempotent semirings, Theoret. Comput. Sci. 534 (2014), 86–100. https://doi.org/10.1016/j.tcs.2014.02.032
[12] A. Deitmar, F1-schemes and toric varieties, Beitr. Algebra Geom. 49(2) (2008), 517–525.
[13] M. Droste, W. Kuich and H. Vogler (Eds.), Handbook of Weighted Automata, Springer, Berlin, 2009.
[14] A. Gathmann, Tropical algebraic geometry, Jahresbericht der Deutschen Mathematiker-Vereinigung 108(1) (2006), 3–32.
[15] J. S. Golan, Semirings and Their Applications, Kluwer Academic Publishers, 1999.
[16] M. Gondran and M. Minoux, Graphs, Dioids and Semirings, Springer, New York, 2008.
[17] U. Hebisch and H. J. Weinert, Semirings: Algebraic Theory and Applications in Computer Science, World Scientific, 1993.
[18] S. N. Il’in, On the homological classification of semirings, J. Math. Sci. (N.Y.) 256(2) (2021), 125–143. https://doi.org/10.1007/s10958-021-05423-1
[19] I. Itenberg, G. Mikhalkin and E. Shustin, Tropical Algebraic Geometry, Birkhäuser Verlag, Basel, 2009.
[20] Z. Izhakian and L. Rowen, Supertropical algebra, Adv. Math. 225(4) (2010), 2222–2286. https://doi.org/10.1016/j.aim.2010.04.007
[21] Z. Izhakian, J. Rhodes and B. Steinberg, Representation theory of finite semigroups over semirings, J. Algebra 336 (2011), 139–157. https://doi.org/10.1016/j.jalgebra.2011.02.048
[22] D. Joó and K. Mincheva, On the dimension of polynomial semirings, J. Algebra 507 (2018), 103–119. https://doi.org/10.1016/j.jalgebra.2018.04.007
[23] D. Joó and K. Mincheva, Prime congruences of additively idempotent semirings and a Nullstellensatz for tropical polynomials, Selecta Math. (N.S.) 24 (2018), 2207–2233. https://doi.org/10.1007/s00029-017-0322-x
[24] Y. Katsov and T. G. Nam, On radicals of semirings and related problems, Comm. Algebra 42 (2014), 5065–5099. https://doi.org/10.1080/00927872.2013.833208
[25] P. Lescot, Absolute algebra II - Ideals and spectra, J. Pure Appl. Algebra 215 (2011), 1782–1790. https://doi.org/10.1016/j.jpaa.2010.10.019
[26] P. Lescot, Absolute algebra III - The saturated spectrum J. Pure Appl. Algebra 216 (2012), 1004–1015. https://doi.org/10.1016/j.jpaa.2011.10.031
[27] G. L. Litvinov, The Maslov dequantization, and idempotent and tropical mathematics: a brief introduction, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 326 (2005), 145–182 (Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 13); translation in: J. Math. Sci. (N.Y.) 140(3) (2007), 426–444.
[28] L. H. Mai and N. X. Tuyen, Some remarks on the Jacobson radical types of semirings and related problems, Vietnam J. Math. 45 (2017), 493–506. https://doi.org/10.1007/s10013-016-0226-7
[29] S. S. Mitchell and P. B. Fenoglio, Congruence-free commutative semirings, Semigroup Forum 31 (1988), 79–91. https://doi.org/10.1007/BF02573125
[30] M. P. Schützenberger, On the definition of a family of automata, Information and Control 4 (1961), 245–270. https://doi.org/10.1016/S0019-9958(61)80020-X
[31] M. K. Sen and A. K. Bhuniya, On semirings whose additive reduct is a semilattice, Semigroup Forum 82 (2011), 131–140. https://doi.org/10.1007/s00233-010-9271-9
[32] D. Wilding, M. Johnson and M. Kambites, Exact rings and semirings, J. Algebra 388 (2013), 324–337. https://doi.org/10.1016/j.jalgebra.2013.05.005
Login:
