Czechoslovak Mathematical Journal, online first, 7 pp.

Skew inverse power series rings over a ring with projective socle

Kamal Paykan

Received December 11, 2015.   First published March 20, 2017.

Kamal Paykan, Department of Basic Sciences, Garmsār Branch, Islamic Azad University, 3581631167 Garmsār, Iran, e-mail: k.paykan@gmail.com, k.paykan@modares.ac.ir

Abstract: A ring $R$ is called a right $\rm PS$-ring if its socle, ${\rm Soc}(R_R )$, is projective. Nicholson and Watters have shown that if $R$ is a right $\rm PS$-ring, then so are the polynomial ring $R[x]$ and power series ring $R[[x]]$. In this paper, it is proved that, under suitable conditions, if $R$ has a (flat) projective socle, then so does the skew inverse power series ring $R[[x^{-1};\alpha, \delta]]$ and the skew polynomial ring $R[x;\alpha, \delta]$, where $R$ is an associative ring equipped with an automorphism $\alpha$ and an $\alpha$-derivation $\delta$. Our results extend and unify many existing results. Examples to illustrate and delimit the theory are provided.

Keywords: skew inverse power series ring; skew polynomial ring; annihilator; projective socle ring; flat socle ring

Classification (MSC 2010): 16W60, 16W70, 16S36, 16P40

DOI: 10.21136/CMJ.2017.0672-15

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References:
  [1] E. P. Armendariz: A note on extensions of Baer and p.p.-rings. J. Aust. Math. Soc. 18 (1974), 470-473. DOI 10.1017/S1446788700029190 | MR 0366979 | Zbl 0292.16009
  [2] K. R. Goodearl: Centralizers in differential, pseudo-differential, and fractional differential operator rings. Rocky Mt. J. Math. 13 (1983), 573-618. DOI 10.1216/RMJ-1983-13-4-573 | MR 0724420 | Zbl 0532.16002
  [3] R. Gordon: Rings in which minimal left ideals are projective. Pac. J. Math. 31 (1969), 679-692. DOI 10.2140/pjm.1969.31.679 | MR 0265404 | Zbl 0188.08402
  [4] E. Hashemi, A. Moussavi: Polynomial extensions of quasi-Baer rings. Acta Math. Hung. 107 (2005), 207-224. DOI 10.1007/s10474-005-0191-1 | MR 2148584 | Zbl 1081.16032
  [5] I. Kaplansky: Rings of Operators. Mathematics Lecture Note Series, W. A. Benjamin, New York (1968). MR 0244778 | Zbl 0174.18503
  [6] C. O. Kim, H. K. Kim, S. H. Jang: A study on quasi-duo rings. Bull. Korean Math. Soc. 36 (1999), 579-588. MR 1722187 | Zbl 0938.16002
  [7] J. Krempa: Some examples of reduced rings. Algebra Colloq. 3 (1996), 289-300. MR 1422968 | Zbl 0859.16019
  [8] T. Y. Lam, A. S. Dugas: Quasi-duo rings and stable range descent. J. Pure Appl. Algebra 195 (2005), 243-259. DOI 10.1016/j.jpaa.2004.08.011 | MR 2114274 | Zbl 1071.16003
  [9] A. Leroy, J. Matczuk, E. R. Puczyłowski: Quasi-duo skew polynomial rings. J. Pure Appl. Algebra 212 (2008), 1951-1959. DOI 10.1016/j.jpaa.2008.01.002 | MR 2414695 | Zbl 1143.16024
  [10] E. S. Letzter, L. Wang: Noetherian skew inverse power series rings. Algebr. Represent. Theory 13 (2010), 303-314. DOI 10.1007/s10468-008-9123-4 | MR 2630122 | Zbl 1217.16038
  [11] Z. K. Liu: Rings with flat left socle. Commun. Algebra 23 (1995), 1645-1656. DOI 10.1080/00927879508825301 | MR 1323692 | Zbl 0826.16002
  [12] Z. Liu, F. Li: PS-rings of generalized power series. Commun. Algebra 26 (1998), 2283-2291. DOI 10.1080/00927879808826276 | MR 1626626 | Zbl 0905.16021
  [13] W. K. Nicholson, J. F. Watters: Rings with projective socle. Proc. Am. Math. Soc. 102 (1988), 443-450. DOI 10.2307/2047200 | MR 0928957 | Zbl 0657.16015
  [14] K. Paykan, A. Moussavi: Special properties of differential inverse power series rings. J. Algebra Appl. 15 (2016), Article ID 1650181, 23 pages. DOI 10.1142/S0219498816501814 | MR 3575971 | Zbl 06667896
  [15] K. Paykan, A. Moussavi: Study of skew inverse Laurent series rings. J. Algebra Appl. 16 (2017), Article ID 1750221, 33 pages. DOI 10.1142/s0219498817502218
  [16] R. M. Salem, M. A. Farahat, H. Abd-Elmalk: PS-modules over Ore extensions and skew generalized power series rings. Int. J. Math. Math. Sci. (2015), Article ID 879129, 6 pages. DOI 10.1155/2015/879129 | MR 3332121
  [17] D. A. Tuganbaev: Laurent series rings and pseudo-differential operator rings. J. Math. Sci., New York 128 (2005), 2843-2893. DOI 10.1007/s10958-005-0244-6 | MR 2171557 | Zbl 1122.16033
  [18] Y. Xiao: Rings with flat socles. Proc. Am. Math. Soc. 123 (1995), 2391-2395. DOI 10.2307/2161264 | MR 1254860 | Zbl 0835.16002
  [19] W. Xue: Modules with projective socles. Riv. Mat. Univ. Parma, V. Ser. 1 (1992), 311-315. MR 1230620 | Zbl 0806.16004
  [20] H.-P. Yu: On quasi-duo rings. Glasg. Math. J. 37 (1995), 21-31. DOI 10.1017/S0017089500030342 | MR 1316960 | Zbl 0819.16001


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