Ukrainian Mathematical Journal

, Volume 67, Issue 2, pp 243–266 | Cite as

Arithmetic of Semigroup Semirings

  • V. Ponomarenko

We define semigroup semirings by analogy with group rings and semigroup rings. We study the arithmetic properties and determine sufficient conditions under which a semigroup semiring is atomic, has finite factorization, or has bounded factorization. We also present a semigroup-semiring analog (although not a generalization) of the Gauss lemma on primitive polynomials.


Group Ring Semigroup Forum Unique Factorization Great Common Divisor Commutative Semigroup 
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  1. 1.
    D. D. Anderson, “GCD domains, Gauss’ lemma, and contents of polynomials, non-Noetherian commutative ring theory,” Math. Appl., 520, 1–31 (2000).Google Scholar
  2. 2.
    P. Cesarz, S. T. Chapman, S. McAdam, and G. J. Schaeffer, “Elastic properties of some semirings defined by positive systems,” Commut. Algebra Appl., Walter de Gruyter, Berlin (2009), pp. 89–101.Google Scholar
  3. 3.
    Ch. Ch. Cheng and R. W. Wong, “Hereditary monoid rings,” Amer. J. Math., 104, No. 5, 935–942 (1982).zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    G. Duchamp and J.-Y. Thibon, “Théorèmes de transfert pour les polynômes partiellement commutatifs,” Theor. and Comput. Sci., 57, No. 2-3, 239–249 (1988).zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    P. Gallagher, “In the finite and nonfinite generation of finitary power semigroups,” Semigroup Forum, 71, No. 3, 481–494 (2005).zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    A. Geroldinger and F. Halter-Koch, “Non-unique factorizations,” Pure and Appl. Math., Chapman & Hall/CRC, Boca Raton, FL 278 (2006).Google Scholar
  7. 7.
    A. Giambruno, C. P. Milies, and S. K. Sehgal (eds.), “Groups, rings and group rings,” Contemp. Math., Amer. Math. Soc., Providence, RI, 499 (2009).Google Scholar
  8. 8.
    R. Gilmer, “Commutative semigroup rings,” Chicago Lect. Math., Univ. Chicago Press, Chicago, IL (1984).Google Scholar
  9. 9.
    J. S. Golan, “The theory of semirings with applications in mathematics and theoretical computer science,” Pitman Monogr. Surv., Pure Appl. Math., Longman Sci. & Techn., Harlow, 54 (1992).Google Scholar
  10. 10.
    J. S. Golan, “Semirings and affine equations over them: theory and applications,” Math. App., 556 (2003).Google Scholar
  11. 11.
    F. Halter-Koch, “Ideal systems,” Monogr. Textbooks, Pure Appl. Math., Marcel Dekker, New York, 211 (1998).Google Scholar
  12. 12.
    G. Révész, “When is a total ordering of a semigroup a well-ordering?,” Semigroup Forum, 41, No. 1, 123–126 (1990).zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    S. Schwarz, “Powers of subsets in a finite semigroup,” Semigroup Forum, 51, No. 1, 1–22 (1995).zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Ch. E. van de Woestijne, “Factors of disconnected graphs and polynomials with nonnegative integer coefficients” (to appear).Google Scholar
  15. 15.
    H. J. Weinert, “On 0-simple semirings, semigroup semirings and two kinds of division semirings,” Semigroup Forum, 28, No. 1-3, 313–333 (1984).zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • V. Ponomarenko
    • 1
  1. 1.San Diego State UniversitySan DiegoUSA

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