Yıl: 2020 Cilt: 24 Sayı: 4 Sayfa Aralığı: 782 - 790 Metin Dili: İngilizce DOI: 10.16984/saufenbilder.733935 İndeks Tarihi: 21-12-2021

On Idempotent Units in Commutative Group Rings

Öz:
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 ଶ = 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 ೒∈௜ௗ(ோ) : ∑೒∈௜ௗ(ோ) 𝑟௚ = 1 and 𝑟௚𝑟௛ = 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. 𝑉൫𝑅(𝐺 × 𝐻)൯ = 𝑖𝑑(𝑅(𝐺 × 𝐻)), 𝑖𝑖. 𝑉൫𝑅(𝐺 × 𝐻)൯ = 𝐺 × 𝑖𝑑(𝑅𝐻), 𝑖𝑖𝑖. 𝑉൫𝑅(𝐺 × 𝐻)൯ = 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]. Keywords: idempotent, unit, group ring, commutative
Anahtar Kelime:

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık
  • [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., vol. 1(103), pp. 31-34, 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.
APA kusmus o (2020). On Idempotent Units in Commutative Group Rings. , 782 - 790. 10.16984/saufenbilder.733935
Chicago kusmus omer On Idempotent Units in Commutative Group Rings. (2020): 782 - 790. 10.16984/saufenbilder.733935
MLA kusmus omer On Idempotent Units in Commutative Group Rings. , 2020, ss.782 - 790. 10.16984/saufenbilder.733935
AMA kusmus o On Idempotent Units in Commutative Group Rings. . 2020; 782 - 790. 10.16984/saufenbilder.733935
Vancouver kusmus o On Idempotent Units in Commutative Group Rings. . 2020; 782 - 790. 10.16984/saufenbilder.733935
IEEE kusmus o "On Idempotent Units in Commutative Group Rings." , ss.782 - 790, 2020. 10.16984/saufenbilder.733935
ISNAD kusmus, omer. "On Idempotent Units in Commutative Group Rings". (2020), 782-790. https://doi.org/10.16984/saufenbilder.733935
APA kusmus o (2020). On Idempotent Units in Commutative Group Rings. Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 24(4), 782 - 790. 10.16984/saufenbilder.733935
Chicago kusmus omer On Idempotent Units in Commutative Group Rings. Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi 24, no.4 (2020): 782 - 790. 10.16984/saufenbilder.733935
MLA kusmus omer On Idempotent Units in Commutative Group Rings. Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol.24, no.4, 2020, ss.782 - 790. 10.16984/saufenbilder.733935
AMA kusmus o On Idempotent Units in Commutative Group Rings. Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi. 2020; 24(4): 782 - 790. 10.16984/saufenbilder.733935
Vancouver kusmus o On Idempotent Units in Commutative Group Rings. Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi. 2020; 24(4): 782 - 790. 10.16984/saufenbilder.733935
IEEE kusmus o "On Idempotent Units in Commutative Group Rings." Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 24, ss.782 - 790, 2020. 10.16984/saufenbilder.733935
ISNAD kusmus, omer. "On Idempotent Units in Commutative Group Rings". Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi 24/4 (2020), 782-790. https://doi.org/10.16984/saufenbilder.733935