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.
Similar content being viewed by others
References
D. D. Anderson, “GCD domains, Gauss’ lemma, and contents of polynomials, non-Noetherian commutative ring theory,” Math. Appl., 520, 1–31 (2000).
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.
Ch. Ch. Cheng and R. W. Wong, “Hereditary monoid rings,” Amer. J. Math., 104, No. 5, 935–942 (1982).
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).
P. Gallagher, “In the finite and nonfinite generation of finitary power semigroups,” Semigroup Forum, 71, No. 3, 481–494 (2005).
A. Geroldinger and F. Halter-Koch, “Non-unique factorizations,” Pure and Appl. Math., Chapman & Hall/CRC, Boca Raton, FL 278 (2006).
A. Giambruno, C. P. Milies, and S. K. Sehgal (eds.), “Groups, rings and group rings,” Contemp. Math., Amer. Math. Soc., Providence, RI, 499 (2009).
R. Gilmer, “Commutative semigroup rings,” Chicago Lect. Math., Univ. Chicago Press, Chicago, IL (1984).
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).
J. S. Golan, “Semirings and affine equations over them: theory and applications,” Math. App., 556 (2003).
F. Halter-Koch, “Ideal systems,” Monogr. Textbooks, Pure Appl. Math., Marcel Dekker, New York, 211 (1998).
G. Révész, “When is a total ordering of a semigroup a well-ordering?,” Semigroup Forum, 41, No. 1, 123–126 (1990).
S. Schwarz, “Powers of subsets in a finite semigroup,” Semigroup Forum, 51, No. 1, 1–22 (1995).
Ch. E. van de Woestijne, “Factors of disconnected graphs and polynomials with nonnegative integer coefficients” (to appear).
H. J. Weinert, “On 0-simple semirings, semigroup semirings and two kinds of division semirings,” Semigroup Forum, 28, No. 1-3, 313–333 (1984).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 2, pp. 213–229, February, 2015.
Rights and permissions
About this article
Cite this article
Ponomarenko, V. Arithmetic of Semigroup Semirings. Ukr Math J 67, 243–266 (2015). https://doi.org/10.1007/s11253-015-1077-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-015-1077-1