Taiwanese Journal of Mathematics

Skew Generalized Power Series Rings and the McCoy Property

Masoome Zahiri, Rasul Mohammadi, Abdollah Alhevaz, and Ebrahim Hashemi

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Abstract

Given a ring $R$, a strictly totally ordered monoid $(S,\preceq)$ and a monoid homomorphism $\omega \colon S \to \operatorname{End}(R)$, one can construct the skew generalized power series ring $R[[S,\omega,\preceq]]$, consisting all of the functions from a monoid $S$ to a coefficient ring $R$ whose support is artinian and narrow, where the addition is pointwise, and the multiplication is given by convolution twisted by an action $\omega$ of the monoid $S$ on the ring $R$. In this paper, we consider the problem of determining some annihilator and zero-divisor properties of the skew generalized power series ring $R[[S,\omega,\preceq]]$ over an associative non-commutative ring $R$. Providing many examples, we investigate relations between McCoy property of skew generalized power series ring, namely $(S,\omega)$-McCoy property, and other standard ring-theoretic properties. We show that if $R$ is a local ring such that its Jacobson radical $J(R)$ is nilpotent, then $R$ is $(S,\omega)$-McCoy. Also if $R$ is a semicommutative semiregular ring such that $J(R)$ is nilpotent, then $R$ is $(S,\omega)$-McCoy ring.

Article information

Source
Taiwanese J. Math., Volume 23, Number 1 (2019), 63-85.

Dates
Received: 7 January 2018
Revised: 1 July 2018
Accepted: 6 August 2018
First available in Project Euclid: 13 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1534125615

Digital Object Identifier
doi:10.11650/tjm/180805

Mathematical Reviews number (MathSciNet)
MR3909990

Zentralblatt MATH identifier
07021718

Subjects
Primary: 16S35: Twisted and skew group rings, crossed products
Secondary: 16N40: Nil and nilpotent radicals, sets, ideals, rings 06F05: Ordered semigroups and monoids [See also 20Mxx]

Keywords
skew generalized power series ring $(S,\omega)$-McCoy ring strictly ordered monoid unique product monoid reversible ring semi-regular ring

Citation

Zahiri, Masoome; Mohammadi, Rasul; Alhevaz, Abdollah; Hashemi, Ebrahim. Skew Generalized Power Series Rings and the McCoy Property. Taiwanese J. Math. 23 (2019), no. 1, 63--85. doi:10.11650/tjm/180805. https://projecteuclid.org/euclid.twjm/1534125615


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References

  • A. Alhevaz and E. Hashemi, An alternative perspective on skew generalized power series rings, Mediterr. J. Math. 13 (2016), no. 6, 4723–4744.
  • A. Alhevaz and D. Kiani, McCoy property of skew Laurent polynomials and power series rings, J. Algebra Appl. 13 (2014), no. 2, 1350083, 23 pp.
  • M. Başer, T. K. Kwak and Y. Lee, The McCoy condition on skew polynomial rings, Comm. Algebra 37 (2009), no. 11, 4026–4037.
  • H. E. Bell, Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc. 2 (1970), 363–368.
  • V. Camillo and P. P. Nielsen, McCoy rings and zero-divisors, J. Pure Appl. Algebra 212 (2008), no. 3, 599–615.
  • P. M. Cohn, Reversible rings, Bull. London Math. Soc. 31 (1999), no. 6, 641–648.
  • G. A. Elliott and P. Ribenboim, Fields of generalized power series, Arch. Math. (Basel) 54 (1990), no. 4, 365–371.
  • E. H. Feller, Properties of primary noncommutative rings, Trans. Amer. Math. Soc. 89 (1958), 79–91.
  • D. E. Fields, Zero divisors and nilpotent elements in power series rings, Proc. Amer. Math. Soc. 27 (1971), 427–433.
  • J. M. Habeb, A note on zero commutative and duo rings, Math. J. Okayama Univ. 32 (1990), 73–76.
  • E. Hashemi, McCoy rings relative to a monoid, Comm. Algebra 38 (2010), no. 3, 1075–1083.
  • E. Hashemi and A. Moussavi, Polynomial extensions of quasi-Baer rings, Acta Math. Hungar. 107 (2005), no. 3, 207–224.
  • N. K. Kim and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra 185 (2003), no. 1-3, 207–223.
  • T. Y. Lam, A First Course in Noncommutative Rings, Second edition, Graduate Texts in Mathematics 131, Springer-Verlag, New York, 2001.
  • G. Marks, R. Mazurek and M. Ziembowski, A new class of unique product monoids with applications to ring theory, Semigroup Forum 78 (2009), no. 2, 210–225.
  • ––––, A unified approach to various generalizations of Armendariz rings, Bull. Aust. Math. Soc. 81 (2010), no. 3, 361–397.
  • G. Mason, Reflexive ideals, Comm. Algebra 9 (1981), no. 17, 1709–1724.
  • R. Mazurek and M. Ziembowski, On von Neumann regular rings of skew generalized power series, Comm. Algebra 36 (2008), no. 5, 1855–1868.
  • ––––, On right McCoy rings and right McCoy rings relative to u.p.-monoids, Commun. Contemp. Math. 17 (2015), no. 4, 1550049, 10 pp.
  • N. H. McCoy, Annihilators in polynomial rings, Amer. Math. Monthly 64 (1957), 28–29.
  • R. Mohammadi, A. Moussavi and M. Zahiri, On annihilations of ideals in skew monoid rings, J. Korean Math. Soc. 53 (2016), no. 2, 381–401.
  • P. P. Nielsen, Semi-commutativity and the McCoy condition, J. Algebra 298 (2006), no. 1, 134–141.
  • J. Okniński, Semigroup Algebras, Monographs and Textbooks in Pure and Applied Mathematics 138, Marcel Dekker, New York, 1991.
  • D. S. Passman, The Algebraic Structure of Group Rings, Pure and Applied Mathematics, Wiley-Interscience, New York, 1977.
  • P. Ribenboim, Special properties of generalized power series, J. Algebra 173 (1995), no. 3, 566–586.
  • ––––, Semisimple rings and von Neumann regular rings of generalized power series, J. Algebra 198 (1997), no. 2, 327–338.
  • A. A. Tuganbaev, Semidistributive Modules and Rings, Mathematics and its Applications 449, Kluwer Academic Publishers, Dordrecht, 1998.
  • S. Yang, X. Song and Z. Liu, Power-serieswise McCoy rings, Algebra Colloq. 18 (2011), no. 2, 301–310.
  • M. Zahiri, A. Moussavi and R. Mohammadi, On annihilator ideals in skew polynomial rings, Bull. Iranian Math. Soc. 43 (2017), no. 5, 1017–1036.