Semigroup Forum

, 78:210

A new class of unique product monoids with applications to ring theory

Research article

Abstract

We introduce a class of ordered monoids defined by the existence of certain “unique products” with respect to artinian and narrow subsets of the monoid. The logical relationships between this and other significant classes of monoids are explicated with several examples. We conclude with results on skew generalized power series rings. The new class of monoids provides the appropriate setting for obtaining results on reduced rings and domains of skew generalized power series, and on analogues of Armendariz rings.

Keywords

Artinian narrow unique product monoid Unique product monoid Skew generalized power series ring Reduced ring Armendariz ring 

References

  1. 1.
    Armendariz, E.P.: A note on extensions of Baer and P.P.-rings. J. Aust. Math. Soc. 18, 470–473 (1974) MATHMathSciNetGoogle Scholar
  2. 2.
    Birkenmeier, G.F., Park, J.P.: Triangular matrix representations of ring extensions. J. Algebra 265(2), 457–477 (2003) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Groenewald, N.J.: A note on extensions of Bear and P.P.-rings. Publ. Inst. Math. (Beograd) (N.S.) 34(48), 71–72 (1983) MathSciNetGoogle Scholar
  4. 4.
    Gustedt, J.: Well-quasi-ordering finite posets and formal languages. J. Comb. Theory Ser. B 65(1), 111–124 (1995) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Higman, G.: Ordering by divisibility in abstract algebras. Proc. Lond. Math. Soc. 2(3), 326–336 (1952) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Hong, C.Y., Kim, N.K., Kwak, T.K.: Ore extensions of Baer and p.p.-rings. J. Pure Appl. Algebra 151(3), 215–226 (2000) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hong, C.Y., Kim, N.K., Kwak, T.K.: On skew Armendariz rings. Commun. Algebra 31(1), 103–122 (2003) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kim, N.K., Lee, K.H., Lee, Y.: Power series rings satisfying a zero divisor property. Commun. Algebra 34(6), 2205–2218 (2006) MATHCrossRefGoogle Scholar
  9. 9.
    Krempa, J.: Some examples of reduced rings. Algebra Colloq. 3(4), 289–300 (1996) MATHMathSciNetGoogle Scholar
  10. 10.
    Kruskal, J.B.: The theory of well-quasi-ordering: A frequently discovered concept. J. Comb. Theory Ser. A 13, 297–305 (1972) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lam, T.Y.: A First Course in Noncommutative Rings. Graduate Texts in Math., vol. 131. Springer, New York (1991) MATHGoogle Scholar
  12. 12.
    Liu, Z.: Armendariz rings relative to a monoid. Commun. Algebra 33(3), 649–661 (2005) MATHCrossRefGoogle Scholar
  13. 13.
    Liu, Z.: Special properties of rings of generalized power series. Commun. Algebra 32(8), 3215–3226 (2004) MATHCrossRefGoogle Scholar
  14. 14.
    Marks, G., Mazurek, R., Ziembowski, M.: Unification of some generalizations of Armendariz rings (2008, in preparation) Google Scholar
  15. 15.
    Mazurek, R., Ziembowski, M.: On von Neumann regular rings of skew generalized power series, Commun. Algebra (2008, to appear) Google Scholar
  16. 16.
    Okniński, J.: Semigroup Algebras. Monographs and Textbooks in Pure and Applied Mathematics, vol. 138. Dekker, New York (1991) Google Scholar
  17. 17.
    Passman, D.S.: The Algebraic Structure of Group Rings. Krieger, Melbourne (1985). Reprint of the 1977 original MATHGoogle Scholar
  18. 18.
    Rege, M.B., Chhawchharia, S.: Armendariz rings. Proc. Jpn. Acad. Ser. A Math. Sci. 73(1), 14–17 (1997) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Ribenboim, P.: Noetherian rings of generalized power series. J. Pure Appl. Algebra 79(3), 293–312 (1992) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Ribenboim, P.: Semisimple rings and von Neumann regular rings of generalized power series. J. Algebra 198(2), 327–338 (1997) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Strojnowski, A.: A note on u.p. groups. Commun. Algebra 8(3), 231–234 (1980) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Yan, X.-F.: Special properties of rings of twisted generalized power series. Xibei Shifan Daxue Xuebao Ziran Kexue Ban 43(2), 20–23 (2007) MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Greg Marks
    • 1
  • Ryszard Mazurek
    • 2
  • Michał Ziembowski
  1. 1.Department of Mathematics and Computer ScienceSt. Louis UniversitySt. LouisUSA
  2. 2.Faculty of Computer ScienceBialystok Technical UniversityBiałystokPoland

Personalised recommendations