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Published online by Cambridge University Press:  23 February 2010

Department of Mathematics and Computer Science, St. Louis University, St. Louis, MO 63103, USA (email:
Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15–351 Bialystok, Poland (email:
Maxwell Institute of Sciences, School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK (email:
For correspondence; e-mail:
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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.

Research Article
Copyright © Australian Mathematical Publishing Association Inc. 2010


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.


[1]Anderson, D. D. and Camillo, V., ‘Armendariz rings and Gaussian rings’, Comm. Algebra 26(7) (1998), 22652272.Google Scholar
[2]Annin, S., ‘Associated primes over skew polynomial rings’, Comm. Algebra 30(5) (2002), 25112528.Google Scholar
[3]Armendariz, E. P., ‘A note on extensions of Baer and P.P.-rings’, J. Aust. Math. Soc. 18 (1974), 470473.Google Scholar
[4]Birkenmeier, G. F., Heatherly, H. E. and Lee, E. K., ‘Completely prime ideals and associated radicals’, in: Ring Theory (Granville, OH, 1992), (eds. Jain, S. K. and Rizvi, S. T.) (World Scientific, Singapore, 1993), pp. 102129.Google Scholar
[5]Birkenmeier, G. F. and Park, J. K., ‘Triangular matrix representations of ring extensions’, J. Algebra 265(2) (2003), 457477.Google Scholar
[6]Camillo, V., Nicholson, W. K. and Yousif, M. F., ‘Ikeda–Nakayama rings’, J. Algebra 226 (2000), 10011010.Google Scholar
[7]Camillo, V. and Nielsen, P. P., ‘McCoy rings and zero-divisors’, J. Pure Appl. Algebra 212(3) (2008), 599615.Google Scholar
[8]Chen, W. and Tong, W., ‘A note on skew Armendariz rings’, Comm. Algebra 33 (2005), 11371140.Google Scholar
[9]Farbman, S. P., ‘The unique product property of groups and their amalgamated free products’, J. Algebra 178(3) (1995), 962990.Google Scholar
[10]Fuchs, L. and Shelah, S., ‘Kaplansky’s problem on valuation rings’, Proc. Amer. Math. Soc. 105(1) (1989), 2530.Google Scholar
[11]Hashemi, E. and Moussavi, A., ‘Polynomial extensions of quasi-Baer rings’, Acta Math. Hungar. 107(3) (2005), 207224.Google Scholar
[12]Hirano, Y., ‘On annihilator ideals of a polynomial ring over a noncommutative ring’, J. Pure Appl. Algebra 168 (2002), 4552.Google Scholar
[13]Hong, C. Y., Kim, N. K. and Kwak, T. K., ‘On skew Armendariz rings’, Comm. Algebra 31(1) (2003), 103122.Google Scholar
[14]Huh, C., Kim, H. K. and Lee, Y., ‘Questions on 2-primal rings’, Comm. Algebra 26(2) (1998), 595600.Google Scholar
[15]Huh, C., Kim, Y. and Smoktunowicz, A., ‘Armendariz rings and semicommutative rings’, Comm. Algebra 30(2) (2002), 751761.Google Scholar
[16]Jategaonkar, A., ‘Skew polynomial rings over orders in Artinian rings’, J. Algebra 21 (1972), 5159.Google Scholar
[17]Kim, N. K. and Lee, Y., ‘Armendariz rings and reduced rings’, J. Algebra 223(2) (2000), 477488.Google Scholar
[18]Kim, N. K., Lee, K. H. and Lee, Y., ‘Power series rings satisfying a zero divisor property’, Comm. Algebra 34(6) (2006), 22052218.Google Scholar
[19]Kosan, M. T., ‘The Armendariz module and its application to the Ikeda–Nakayama module’, Int. J. Math. Math. Sci. 2006 (2006), 7. Article ID 35238.Google Scholar
[20]Krempa, J., ‘Some examples of reduced rings’, Algebra Colloq. 3(4) (1996), 289300.Google Scholar
[21]Lam, T. Y., A First Course in Noncommutative Rings, Graduate Texts in Mathematics, 131 (Springer, New York, 1991).Google Scholar
[22]Lee, T.-K. and Wong, T.-L., ‘On Armendariz rings’, Houston J. Math. 29(3) (2003), 583593.Google Scholar
[23]Lee, T.-K. and Zhou, Y., ‘A unified approach to the Armendariz property of polynomial rings and power series rings’, Colloq. Math. 113(1) (2008), 151168.Google Scholar
[24]Liu, Z., ‘Special properties of rings of generalized power series’, Comm. Algebra 32(8) (2004), 32153226.Google Scholar
[25]Liu, Z., ‘Armendariz rings relative to a monoid’, Comm. Algebra 33(3) (2005), 649661.Google Scholar
[26]Marks, G., ‘Direct product and power series formations over 2-primal rings’, in: Advances in Ring Theory, 26, (eds. Jain, S. K. and Rizvi, S. T.) (Birkhäuser, Boston, MA, 1997), pp. 239245.Google Scholar
[27]Marks, G., ‘Reversible and symmetric rings’, J. Pure Appl. Algebra 174(3) (2002), 311318.Google Scholar
[28]Marks, G., ‘A taxonomy of 2-primal rings’, J. Algebra 266(2) (2003), 494520.Google Scholar
[29]Marks, G., Mazurek, R. and Ziembowski, M., ‘A new class of unique product monoids with applications to ring theory’, Semigroup Forum 78(2) (2009), 210225.Google Scholar
[30]Matczuk, J., ‘A characterization of σ-rigid rings’, Comm. Algebra 32(11) (2004), 43334336.Google Scholar
[31]Mazurek, R. and Ziembowski, M., ‘On von Neumann regular rings of skew generalized power series’, Comm. Algebra 36(5) (2008), 18551868.Google Scholar
[32]Moussavi, A. and Hashemi, E., ‘On (α,δ)-skew Armendariz rings’, J. Korean Math. Soc. 42(2) (2005), 353363.Google Scholar
[33]Nagore, C. S. and Satyanarayana, M., ‘On naturally ordered semigroups’, Semigroup Forum 18(2) (1979), 95103.Google Scholar
[34]Nasr-Isfahani, A. R. and Moussavi, A., ‘On classical quotient rings of skew Armendariz rings’, Int. J. Math. Math. Sci. 2007 (2007), 7. Article ID 61549.Google Scholar
[35]Okniński, J., Semigroup Algebras, Monographs and Textbooks in Pure and Applied Mathematics, 138 (Marcel Dekker, New York, 1991).Google Scholar
[36]Puczyłowski, E. R., ‘Questions related to Koethe’s nil ideal problem’, in: Algebra and its Applications, Contemporary Mathematics, 419 (American Mathematical Society, Providence, RI, 2006), pp. 269283.Google Scholar
[37]Rege, M. B. and Chhawchharia, S., ‘Armendariz rings’, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), 1417.Google Scholar