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Year 2020, Volume: 24 Issue: 4, 782 - 790, 01.08.2020
https://doi.org/10.16984/saufenbilder.733935

Abstract

References

  • [1] P. Danchev, “Trivial units in commutative group algebras,” Extr. Math., vol. 23, pp. 49-60, 2008.
  • [2] P. Danchev, “Trivial units in abelian group algebras,” Extr. Math., vol. 24, pp. 47-53, 2009.
  • [3] P. Danchev, “Idempotent units in commutative group rings,” Kochi J. Math, vol. 4, pp. 61-68, 2009.
  • [4] P. Danchev, “Idempotent units of commutative group rings,” Commun. Algebra, vol. 38, pp. 4649-4654, 2010.
  • [5] P. Danchev, “On some idempotent torsion decompositions of normed units in commutative group rings,” J. Calcutta Math. Soc., vol. 6, pp. 31-34, 2010.
  • [6] P. Danchev, “Idempotent-torsion normalized units in abelian group rings,” Bull Calcutta Math. Soc., to appear, 2011.
  • [7] G. Karpilovsky, “On units in commutative group rings,” Arch. Math. (Basel), vol. 38, pp. 420–422, 1982.
  • [8] G. Karpilovsky, “On finite generation of unit groups of commutative group rings,” Arch. Math. (Basel), vol. 40, pp. 503–508, 1983.
  • [9] G. Karpilovsky, “Unit groups of group rings,” Harlow: Longman Sci. and Techn., 1989.
  • [10] G. Karpilovsky, “Units of commutative group algebras,” Expo. Math., vol. 8, pp. 247-287, 1990.
  • [11] W. May, “Group algebras over finitely generated rings,” J. Algebra vol. 39 pp. 483–511, 1976.
  • [12] C. Polcino Milies and S. K. Sehgal, “An introduction to group rings,” Kluwer, North-Holland, Amsterdam, 2002.
  • [13] S. K. Sehgal, “Topics in group rings,” Marcel Dekker, New York, 1978.

On Idempotent Units in Commutative Group Rings

Year 2020, Volume: 24 Issue: 4, 782 - 790, 01.08.2020
https://doi.org/10.16984/saufenbilder.733935

Abstract

Special elements as units, which are defined utilizing idempotent elements, have a very crucial place in a commutative group ring. As a remark, we note that an element is said to be idempotent if r^2=r in a ring. For a group ring RG, idempotent units are defined as finite linear combinations of elements of G over the idempotent elements in R or formally, idempotent units can be stated as of the form id(RG)={∑_(r_g∈id(R))▒〖r_g g〗: ∑_(r_g∈id(R))▒r_g =1 and r_g r_h=0 when g≠h} where id(R) is the set of all idempotent elements [3], [4], [5], [6]. Danchev [3] introduced some necessary and sufficient conditions for all the normalized units are to be idempotent units for groups of orders 2 and 3. In this study, by considering some restrictions, we investigate necessary and sufficient conditions for equalities:
i.V(R(G×H))=id(R(G×H)),
ii.V(R(G×H))=G×id(RH),
iii.V(R(G×H))=id(RG)×H
where G×H is the direct product of groups G and H. Therefore, the study can be seen as a generalization of [3], [4]. Notations mostly follow [12], [13].

References

  • [1] P. Danchev, “Trivial units in commutative group algebras,” Extr. Math., vol. 23, pp. 49-60, 2008.
  • [2] P. Danchev, “Trivial units in abelian group algebras,” Extr. Math., vol. 24, pp. 47-53, 2009.
  • [3] P. Danchev, “Idempotent units in commutative group rings,” Kochi J. Math, vol. 4, pp. 61-68, 2009.
  • [4] P. Danchev, “Idempotent units of commutative group rings,” Commun. Algebra, vol. 38, pp. 4649-4654, 2010.
  • [5] P. Danchev, “On some idempotent torsion decompositions of normed units in commutative group rings,” J. Calcutta Math. Soc., vol. 6, pp. 31-34, 2010.
  • [6] P. Danchev, “Idempotent-torsion normalized units in abelian group rings,” Bull Calcutta Math. Soc., to appear, 2011.
  • [7] G. Karpilovsky, “On units in commutative group rings,” Arch. Math. (Basel), vol. 38, pp. 420–422, 1982.
  • [8] G. Karpilovsky, “On finite generation of unit groups of commutative group rings,” Arch. Math. (Basel), vol. 40, pp. 503–508, 1983.
  • [9] G. Karpilovsky, “Unit groups of group rings,” Harlow: Longman Sci. and Techn., 1989.
  • [10] G. Karpilovsky, “Units of commutative group algebras,” Expo. Math., vol. 8, pp. 247-287, 1990.
  • [11] W. May, “Group algebras over finitely generated rings,” J. Algebra vol. 39 pp. 483–511, 1976.
  • [12] C. Polcino Milies and S. K. Sehgal, “An introduction to group rings,” Kluwer, North-Holland, Amsterdam, 2002.
  • [13] S. K. Sehgal, “Topics in group rings,” Marcel Dekker, New York, 1978.
There are 13 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Ömer Küsmüş 0000-0001-7397-0735

Publication Date August 1, 2020
Submission Date May 7, 2020
Acceptance Date June 10, 2020
Published in Issue Year 2020 Volume: 24 Issue: 4

Cite

APA Küsmüş, Ö. (2020). On Idempotent Units in Commutative Group Rings. Sakarya University Journal of Science, 24(4), 782-790. https://doi.org/10.16984/saufenbilder.733935
AMA Küsmüş Ö. On Idempotent Units in Commutative Group Rings. SAUJS. August 2020;24(4):782-790. doi:10.16984/saufenbilder.733935
Chicago Küsmüş, Ömer. “On Idempotent Units in Commutative Group Rings”. Sakarya University Journal of Science 24, no. 4 (August 2020): 782-90. https://doi.org/10.16984/saufenbilder.733935.
EndNote Küsmüş Ö (August 1, 2020) On Idempotent Units in Commutative Group Rings. Sakarya University Journal of Science 24 4 782–790.
IEEE Ö. Küsmüş, “On Idempotent Units in Commutative Group Rings”, SAUJS, vol. 24, no. 4, pp. 782–790, 2020, doi: 10.16984/saufenbilder.733935.
ISNAD Küsmüş, Ömer. “On Idempotent Units in Commutative Group Rings”. Sakarya University Journal of Science 24/4 (August 2020), 782-790. https://doi.org/10.16984/saufenbilder.733935.
JAMA Küsmüş Ö. On Idempotent Units in Commutative Group Rings. SAUJS. 2020;24:782–790.
MLA Küsmüş, Ömer. “On Idempotent Units in Commutative Group Rings”. Sakarya University Journal of Science, vol. 24, no. 4, 2020, pp. 782-90, doi:10.16984/saufenbilder.733935.
Vancouver Küsmüş Ö. On Idempotent Units in Commutative Group Rings. SAUJS. 2020;24(4):782-90.