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Quasi-Armendariz generalized power series rings

Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P. O. Box 14115-134, Tehran, Iran

E-mail Address: k.paykan@gmail.com

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and

A. Moussavi

Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P. O. Box 14115-134, Tehran, Iran

E-mail Address: moussavi.a@modares.ac.ir; moussavi.a@gmail.com

Corresponding author.

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https://doi.org/10.1142/S0219498816500869Cited by:2 (Source: Crossref)

Abstract

Let $R$ $R$ be a ring, $(S, \leq)$ $(S, \leq)$ a strictly ordered monoid and $ω : S \to End (R)$ $ω : S \to End (R)$ a monoid homomorphism. The skew generalized power series ring $R [[S, ω]]$ $R [[S, ω]]$ 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 initiate the study of the $(S, ω)$ $(S, ω)$ -quasi-Armendariz condition on $R$ $R$ , a generalization of the standard quasi-Armendariz condition from polynomials to skew generalized power series. The class of quasi-Armendariz rings includes semiprime rings, Armendariz rings, right (left) p.q.-Baer rings and right (left) PP rings. The $(S, ω)$ $(S, ω)$ -quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. The $2 \times 2$ $2 \times 2$ formal upper triangular matrix rings of this class are characterized. We conclude some characterizations for a skew generalized power series ring to be semiprime, quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and left AIP. Examples to illustrate and delimit the theory are provided.

Communicated by L. Bokut

Keywords:

AMSC: 16S34, 16S35, 16S36, 16W60, 16D40, 16P99, 16W20, 20M99

References

1. E. P. Armendariz, A note on extensions of Baer and p.p.-rings, J. Austral. Math. Soc. 18 (1974) 470–473. Crossref, Google Scholar
2. M. Baser, F. Kaynarca, T. K. Kwak and Y. Lee, Weak quasi-Armendariz rings, Algebra Colloq. 18(4) (2011) 541–552. Link, Web of Science, Google Scholar
3. M. Baser and T. K. Kwak, Quasi-Armendariz property for skew polynomial rings, Commun. Korean Math. Soc. 26(4) (2011) 557–573. Crossref, Google Scholar
4. H. E. Bell, Near-rings in which each element is a power of itself, Bull. Aust. Math. Soc. 2 (1970) 363–368. Crossref, Google Scholar
5. S. K. Berberian, Baer*-Rings (Springer, Berlin, 1972). Crossref, Google Scholar
6. G. M. Bergman and I. M. Issacs, Rings with fixed-point-free group actions, Proc. London Math. Soc. 27 (1973) 69–87. Crossref, Web of Science, Google Scholar
7. G. F. Birkenmeier, J. Y. Kim and J. K. Park, On quasi-Baer rings, Contemp. Math. 259 (2000) 67–92. Crossref, Google Scholar
8. G. F. Birkenmeier, J. Y. Kim and J. K. Park, Principally quasi-Baer rings, Comm. Algebra 29(2) (2001) 639–660. Crossref, Web of Science, Google Scholar
9. G. F. Birkenmeier, J. Y. Kim and J. K. Park, Polynomial extensions of Baer and quasi-Baer rings, J. Pure Appl. Algebra 159 (2001) 24–42. Crossref, Web of Science, Google Scholar
10. G. F. Birkenmeier, J. Y. Kim and J. K. Park, Right primary and nilary rings and ideals, J. Algebra 378 (2013) 133–152. Crossref, Web of Science, Google Scholar
11. G. F. Birkenmeier and J. K. Park, Triangular matrix representations of ring extensions, J. Algebra 265(2) (2003) 457–477. Crossref, Web of Science, Google Scholar
12. V. Camillo, T. K. Kwak and Y. Lee, Ideal-symmetric and semiprime rings, Comm. Algebra 41 (2013) 4504–4519. Crossref, Web of Science, Google Scholar
13. J. Chen, X. Yang and Y. Zhou, On strongly clean matrix and triangular matrix rings, Comm. Algebra 34 (2006) 3659–3674. Crossref, Web of Science, Google Scholar
14. W. E. Clark, Twisted matrix units semigroup algebras, Duke Math. J. 34 (1967) 417–424. Crossref, Web of Science, Google Scholar
15. P. M. Cohn, A Morita context related to finite automorphism groups of rings, Pacific J. Math. 98(1) (1982) 37–54. Crossref, Web of Science, Google Scholar
16. P. M. Cohn, Free Rings and Their Relations, 2nd edn. (Academic Press, London, 1985). Google Scholar
17. P. M. Cohn, Reversible rings, Bull. London Math. Soc. 31(6) (1999) 641–648. Crossref, Web of Science, Google Scholar
18. I. G. Connell, On the group ring, Canad. J. Math. 15 (1963) 650–685. Crossref, Web of Science, Google Scholar
19. G. A. Elliott and P. Ribenboim, Fields of generalized power series, Arch. Math. 54 (1990) 365–371. Crossref, Web of Science, Google Scholar
20. S. P. Farbman, The unique product property of groups and their amalgamated free products, J. Algebra 178(3) (1995) 962–990. Crossref, Web of Science, Google Scholar
21. D. E. Fields, Zero divisors and nilpotent elements in power series rings, Proc. Amer. Math. Soc. 27(3) (1971) 427–433. Crossref, Web of Science, Google Scholar
22. J. W. Fisher and S. Montgomery, Semiprime skew group rings, J. Algebra 52(1) (1978) 241–247. Crossref, Web of Science, Google Scholar
23. Gilmer and T. Parker, Zero divisors in power series rings, J. Reine Angew. Math. 278/279 (1975) 145–164. Google Scholar
24. E. Hashemi, Quasi-Armendariz rings relative to a monoid, J. Pure Appl. Algebra 2 (2007) 374–382. Crossref, Web of Science, Google Scholar
25. E. Hashemi and A. Moussavi, Polynomial extensions of quasi-Baer rings, Acta Math. Hungar. 107(3) (2005) 207–224. Crossref, Web of Science, Google Scholar
26. Y. Hirano, On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra 168 (2002) 45–52. Crossref, Web of Science, Google Scholar
27. C. Y. Hong, N. K. Kim and T. K. Kwak, Ore extensions of Baer and p.p.-rings, J. Pure Appl. Algebra 151 (2000) 215–226. Crossref, Web of Science, Google Scholar
28. C. Y. Hong, N. K. Kim and Y. Lee, Extensions of McCoy’s theorem, Glasg. Math. J. 52(1) (2010) 155–159. Crossref, Web of Science, Google Scholar
29. C. Y. Hong, N. K. Kim and Y. Lee, Skew polynomial rings over semiprime rings, J. Korean Math. Soc. 47(5) (2010) 879–897. Crossref, Web of Science, Google Scholar
30. D. A. Jordan, Bijective extensions of injective ring endomorphisms, J. London Math. Soc. 25(2) (1982) 435–448. Crossref, Google Scholar
31. J. W. Kerr, The polynomial ring over a Goldie ring need not be a Goldie ring, J. Algebra 134 (1990) 344–352. Crossref, Web of Science, Google Scholar
32. N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000) 477–488. Crossref, Web of Science, Google Scholar
33. J. Krempa, Some examples of reduced rings, Algebra Colloq. 3(4) (1996) 289–300. Google Scholar
34. T. K. Kwak and Y. Lee, Reflexive property of rings, Comm. Algebra 40 (2012) 1576–1594. Crossref, Web of Science, Google Scholar
35. T. Y. Lam, A First Course in Noncommutative Rings, Vol. 131 (Springer, New York, 1991). Crossref, Google Scholar
36. T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, Vol. 89 (Springer, New York, 1999). Crossref, Google Scholar
37. A. Leroy and J. Matczuk, Goldie conditions for Ore extensions over semiprime rings, Algebra Represent. Theory 8(5) (2005) 679–688. Crossref, Web of Science, Google Scholar
38. Z. K. Liu, Endomorphism rings of modules of generalized inverse polynomials, Comm. Algebra 28(2) (2000) 803–814. Crossref, Web of Science, Google Scholar
39. Z. K. Liu, Triangular matrix representations of rings of generalized power series, Acta Math. Sinica (English Series) 22 (2006) 989–998. Crossref, Web of Science, Google Scholar
40. Z. K. Liu and Z. Wenhui, Quasi-Armendariz rings relative to a monoid, Comm. Algebra 36(3) (2008) 928–947. Crossref, Web of Science, Google Scholar
41. Z. Liu and R. Zhao, A generalization of PP-rings and p.q.-Baer rings, Glasg. Math. J. 48(2) (2006) 217–229. Crossref, Web of Science, Google Scholar
42. A. Majidinya, A. Moussavi and K. Paykan, Rings in which the annihilator of an ideal is pure, to appear in Algebra Colloq. Google Scholar
43. G. Marks, R. Mazurek and M. Ziembowski, A new class of unique product monoids with applications to ring theory, Semigroup Forum 78(2) (2009) 210–225. Crossref, Web of Science, Google Scholar
44. G. Marks, R. Mazurek and M. Ziembowski, A unified approach to various generalizations of Armendariz rings, Bull. Aust. Math. Soc. 81 (2010) 361–397. Crossref, Web of Science, Google Scholar
45. G. Mason, Reflexive ideals, Comm. Algebra 9 (1981) 1709–1724. Crossref, Web of Science, Google Scholar
46. R. Mazurek and M. Ziembowski, On Neumann regular rings of skew generalized power series, Comm. Algebra 36(5) (2008) 1855–1868. Crossref, Web of Science, Google Scholar
47. N. H. McCoy, Annihilators in polynomial rings, Amer. Math. Monthly 64 (1957) 28–29. Crossref, Google Scholar
48. S. Montgomery, Outer automorphisms of semi-prime rings, J. London Math. Soc. 18(2) (1978) 209–220. Crossref, Google Scholar
49. A. Moussavi, H. Haj Seyyed Javadi and E. Hashemi, Generalized quasi-Baer rings, Comm. Algebra 33(7) (2005) 2115–2129. Crossref, Web of Science, Google Scholar
50. A. R. Nasr-Isfahani and A. Moussavi, On weakly rigid rings. Glasg. Math. J. 51(3) (2009) 425–440. Crossref, Web of Science, Google Scholar
51. D. S. Passman, The Algebraic Structure of Group Rings (Wiley-Interscience, New York, 1977). Google Scholar
52. K. Paykan, A. Moussavi and M. Zahiri, Special properties of rings of skew generalized power series, Comm. Algebra 42(12) (2014) 5224–5248. Crossref, Web of Science, Google Scholar
53. A. Pollingher and A. Zaks, On Baer and quasi-Baer rings, Duke Math. J. 37 (1970) 127–138. Crossref, Web of Science, Google Scholar
54. M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997) 14–17. Crossref, Web of Science, Google Scholar
55. P. Ribenboim, Special properties of generalized power series, J. Algebra 173 (1995) 566–586. Crossref, Web of Science, Google Scholar
56. P. Ribenboim, Some examples of valued fields, J. Algebra 173 (1995) 668–678. Crossref, Web of Science, Google Scholar
57. P. Ribenboim, Semisimple rings and von Neumann regular rings of generalized power series, J. Algebra 198 (1997) 327–338. Crossref, Web of Science, Google Scholar
58. B. Stenstrom, Rings of Quotients (Springer, 1975). Crossref, Google Scholar
59. H. Tominaga, On s-unital rings, Math. J. Okayama Univ. 18(2) (1976) 117–134. Google Scholar
60. A. A. Tuganbaev, Some ring and module properties of skew Laurent series, in Formal Power Series and Algebraic Combinatorics (Moscow) (Springer, Berlin, 2000), pp. 613–622. Crossref, Google Scholar