Abstract
For an integral domain R and a commutative cancellative monoid M, the ring consisting of all polynomial expressions with coefficients in R and exponents in M is called the monoid ring of M over R. An integral domain R is called atomic if every nonzero nonunit element can be written as a product of irreducibles. In the study of the atomicity of integral domains, the building blocks are the irreducible elements. Thus, tools to prove irreducibility are crucial to study atomicity. In the first part of this paper, we extend Gauss’s Lemma and Eisenstein’s Criterion from polynomial rings to monoid rings. An integral domain R is called half-factorial (or an HFD) if any two factorizations of a nonzero nonunit element of R have the same number of irreducible elements (counting repetitions). In the second part of this paper, we determine which monoid algebras with nonnegative rational exponents are Dedekind domains, Euclidean domains, PIDs, UFDs, and HFDs. As a side result, we characterize the submonoids of 
satisfying a dual notion of half-factoriality known as other-half-factoriality.
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References
Amini, M.: Module amenability for semigroup algebras. Semigroup Forum 69, 243–254 (2004)
Anderson, D.D., Juett, J.R.: Long length functions. J. Algebra 426, 327–343 (2015)
Anderson, D.F., Scherpenisse, D.: Factorization in K[S]. In: Anderson, D.D. (ed.) Factorization in Integral Domains. Lecture Notes in Pure and Applied Mathematics, vol. 189, pp. 45–56. Marcel Dekker, New York (1997)
Anderson, D.D., Anderson, D.F., Zafrullah, M.: Factorizations in integral domains. J. Pure Appl. Algebra 69, 1–19 (1990)
Anderson, D.D., Coykendall, J., Hill, L., Zafrullah, M.: Monoid domain constructions of antimatter domains. Comm. Alg. 35, 3236–3241 (2007)
Assi, A., García-Sánchez, P.A.: Numerical Semigroups and Applications. RSME Springer Series. Springer, New York (2016)
Briales, E., Campillo, A., Marijuán, C., Pisón, P.: Combinatorics and syzygies for semigroup algebras. Coll. Math. 49, 239–256 (1998)
Bruns, W., Gubeladze, J.: Semigroup algebras and discrete geometry. In: Bonavero, L., Biron, M. (eds.) Geometry of Toric Varieties. Séminaires et Congrés, vol. 6, pp. 43–127. Mathematical Society of France, Paris (2002)
Carlitz, L.: A characterization of algebraic number fields with class number two. Proc. Am. Math. Soc. 11, 391–392 (1960)
Chapman, S.T., Gotti, F., Gotti, M.: Factorization invariants of Puiseux monoids generated by geometric sequences. Comm. Alg. 48, 380–396 (2020)
Coykendall, J., Gotti, F.: On the atomicity of monoid algebras. J. Algebra 539, 138–151 (2019)
Coykendall, J., Smith, W.W.: On unique factorization domains. J. Algebra 332, 62–70 (2011)
García-Sánchez, P.A., Rosales, J.C.: Numerical Semigroups. Developments in Mathematics, vol. 20. Springer, New York (2009)
Gilmer, R.: A two-dimensional non-Noetherian factorial ring. Proc. Am. Math. Soc. 44, 25–30 (1974)
Gilmer, R.: Commutative Semigroup Rings. Chicago Lectures in Mathematics. The University of Chicago Press, London (1984)
Gilmer, R., Parker, T.: Divisibility properties of semigroup rings. Mich. Math. J. 21, 65–86 (1974)
Gilmer, R., Parker, T.: Nilpotent elements of commutative semigroup rings. Mich. Math. J. 22, 97–108 (1975)
Gotti, F.: On the atomic structure of Puiseux monoids. J. Algebra Appl. 16, 1750126 (2017)
Gotti, F.: Puiseux monoids and transfer homomorphisms. J. Algebra 516, 95–114 (2018)
Gotti, F.: Increasing positive monoids of ordered fields are FF-monoids. J. Algebra 518, 40–56 (2019)
Gotti, F.: Systems of sets of lengths of Puiseux monoids. J. Pure Appl. Algebra 223, 1856–1868 (2019)
Gotti, F., O’Neill, C.: The elasticity of Puiseux monoids. J. Commut. Algebra (to appear). https://projecteuclid.org/euclid.jca/1523433696
Grams, A.: Atomic domains and the ascending chain condition for principal ideals. Math. Proc. Cambridge Philos. Soc. 75, 321–329 (1974)
Grillet, P.A.: Commutative Semigroups. Advances in Mathematics, vol. 2. Kluwer Academic Publishers, Boston (2001)
Kim, H.: Factorization in monoid domains. Commun. Algebra 29, 1853–1869 (2001)
Zaks, A.: Half-factorial domains. Bull. Am. Math. Soc. 82, 721–723 (1976)
Acknowledgements
While working on this paper, the author was supported by the NSF-AGEP Fellowship and the UC Dissertation Year Fellowship. The author would like to thank an anonymous referee, whose suggestions help to simplify and improve the initially-submitted version of this paper.
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Gotti, F. (2020). Irreducibility and Factorizations in Monoid Rings. In: Barucci, V., Chapman, S., D'Anna, M., Fröberg, R. (eds) Numerical Semigroups . Springer INdAM Series, vol 40. Springer, Cham. https://doi.org/10.1007/978-3-030-40822-0_9
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