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A UNIFIED APPROACH TO VARIOUS GENERALIZATIONS OF ARMENDARIZ RINGS

  • GREG MARKS (a1), RYSZARD MAZUREK (a2) and MICHAŁ ZIEMBOWSKI (a3)
Abstract
Abstract

Let R be a ring, S a strictly ordered monoid, and ω:SEnd(R) a monoid homomorphism. The skew generalized power series ring R[[S,ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. We study the (S,ω)-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. We resolve the structure of (S,ω)-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be (S,ω)-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the aforementioned special cases. As particular cases of our general results we obtain several new theorems on the Armendariz condition; for example, left uniserial rings are Armendariz. We also characterize when a skew generalized power series ring is reduced or semicommutative, and we obtain partial characterizations for it to be reversible or 2-primal.

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Copyright
Corresponding author
For correspondence; e-mail: marks@slu.edu
Footnotes
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The second author was supported by Bialystok University of Technology grant W/WI/7/08, MNiSW grant N N201 268435, and KBN grant 1 P03A 032 27.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[21] T. Y. Lam , A First Course in Noncommutative Rings, Graduate Texts in Mathematics, 131 (Springer, New York, 1991).

[26] G. Marks , ‘Direct product and power series formations over 2-primal rings’, in: Advances in Ring Theory, 26, (eds. S. K. Jain and S. T. Rizvi ) (Birkhäuser, Boston, MA, 1997), pp. 239245.

[36] E. R. Puczyłowski , ‘Questions related to Koethe’s nil ideal problem’, in: Algebra and its Applications, Contemporary Mathematics, 419 (American Mathematical Society, Providence, RI, 2006), pp. 269283.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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