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A Generalization of G-Nilpotent Units in Commutative Group Rings to Direct Product Groups

Year 2023, Volume: 28 Issue: 1, 8 - 18, 30.04.2023
https://doi.org/10.53433/yyufbed.1097581

Abstract

Let V(RG) denote the normalized unit group of the group ring RG of a group G over a ring R. The concept of G-nilpotent unit in a commutative group ring has been defined in (Danchev, 2012). In this study, some necessary and sufficient conditions for a normalized unit group in a commutative group ring of a direct product group G×H to consist only of G×H-nilpotent units have been given and especially some results which are related to groups G×C_3 and G×C_4 have been introduced where C_3 and C_4 are cyclic groups of orders 3 and 4 respectively. In this context, we can say that the paper extends the results in (Danchev, 2012). At the end, an open problem is served as a future work.

References

  • Bourbaki, N. (1989). Elements of Mathematics, Commutative Algebra. Berlin, Germany: Springer.
  • Danchev, P. (2008). Trivial units in commutative group algebras. Extracta mathematicae, 23(1), 49-60.
  • Danchev, P. (2009). Trivial units in abelian group algebras. Extracta mathematicae, 24(1), 47-53.
  • Danchev, P. (2010). Idempotent units of commutative group rings. Communications in Algebra, 38(12), 4649-4654. doi:10.1080/00927871003742842
  • Danchev, P. (2012). G-nilpotent units in Abelian group rings. Commentationes Mathematicae Universitatis Carolinae, 53(2), 179-187.
  • Görentaş, N. (1999). A characterization of idempotents and idempotent generators of QS_3. Bulletion of pure and Applied Sciences, 2(18), 289-292.
  • Görentaş, N. (2020). A note on simple trinomial units in U_1 (ZC_p). Turkish Journal of Mathematics, 44(5), 1783-1791. doi:10.3906/mat-2003-63
  • Karpilovsky, G. (1982). On units in commutative group rings. Archiv der Mathematik, 38, 420–422. doi:10.1007/BF01304809
  • Küsmüş, Ö. (2019,Aralık). Nilpotent, idempotent and units in group rings. (PhD), Yuzuncu Yıl University, Institute of Natural and Applied Science Van, Turkey.
  • Küsmüş, Ö. (2020). On idempotent units in commutative group rings. Sakarya University Journal of Science, 24(4), 782-790. doi:10.16984/saufenbilder.733935
  • Li, Y. (1998). Units of Z(G×C_2). Quaestiones Mathematicae, 21(3-4), 201-218. doi:10.1080/16073606.1998.9632041
  • May, W. (1976). Group algebras over finitely generated rings. Journal of Algebra, 39(2), 483–511. doi:10.1016/0021-8693(76)90049-1
  • Milies, C. P., & Sehgal, S. K. (2002). An Introduction to Group Rings. Amsterdam, North-Holland: Kluwer.
  • Sehgal, S. K. (1978). Topics in group rings. New York, US: Marcel Dekker.

Değişmeli Grup Halkalarında G-Nilpotent Birimsel Elemanların Direkt Çarpım Gruplarına Bir Genellemesi

Year 2023, Volume: 28 Issue: 1, 8 - 18, 30.04.2023
https://doi.org/10.53433/yyufbed.1097581

Abstract

V(RG), bir R halkası üzerindeki bir G grubunun RG grup halkasının normalleştirilmiş birim grubunu göstersin. Değişmeli bir grup halkasındaki G-nilpotent birimsel kavramı (Danchev, 2012)'de tanımlanmıştır. Bu çalışmada da, bir G×H direkt çarpım grubunun değişmeli grup halkasında normallenmiş birimsel elemanlar grubunun sadece G×H-nilpotent birimsel elemanlardan oluşabilmesi için bazı gerek ve yeter şartlar verilmiştir. Ayrıca özel olarak G×C_3 ve G×C_4 gruplarına dair bazı sonuçlar sunulmuştur ki burada C_3 ve C_4 sırasıyla 3 ve 4 mertebeli devirli gruplardır. Bu bağlamda, makale (Danchev, 2012)’deki sonuçları genişletir diyebiliriz. Sonunda, gelecek çalışma için açık problem sunulmuştur.

References

  • Bourbaki, N. (1989). Elements of Mathematics, Commutative Algebra. Berlin, Germany: Springer.
  • Danchev, P. (2008). Trivial units in commutative group algebras. Extracta mathematicae, 23(1), 49-60.
  • Danchev, P. (2009). Trivial units in abelian group algebras. Extracta mathematicae, 24(1), 47-53.
  • Danchev, P. (2010). Idempotent units of commutative group rings. Communications in Algebra, 38(12), 4649-4654. doi:10.1080/00927871003742842
  • Danchev, P. (2012). G-nilpotent units in Abelian group rings. Commentationes Mathematicae Universitatis Carolinae, 53(2), 179-187.
  • Görentaş, N. (1999). A characterization of idempotents and idempotent generators of QS_3. Bulletion of pure and Applied Sciences, 2(18), 289-292.
  • Görentaş, N. (2020). A note on simple trinomial units in U_1 (ZC_p). Turkish Journal of Mathematics, 44(5), 1783-1791. doi:10.3906/mat-2003-63
  • Karpilovsky, G. (1982). On units in commutative group rings. Archiv der Mathematik, 38, 420–422. doi:10.1007/BF01304809
  • Küsmüş, Ö. (2019,Aralık). Nilpotent, idempotent and units in group rings. (PhD), Yuzuncu Yıl University, Institute of Natural and Applied Science Van, Turkey.
  • Küsmüş, Ö. (2020). On idempotent units in commutative group rings. Sakarya University Journal of Science, 24(4), 782-790. doi:10.16984/saufenbilder.733935
  • Li, Y. (1998). Units of Z(G×C_2). Quaestiones Mathematicae, 21(3-4), 201-218. doi:10.1080/16073606.1998.9632041
  • May, W. (1976). Group algebras over finitely generated rings. Journal of Algebra, 39(2), 483–511. doi:10.1016/0021-8693(76)90049-1
  • Milies, C. P., & Sehgal, S. K. (2002). An Introduction to Group Rings. Amsterdam, North-Holland: Kluwer.
  • Sehgal, S. K. (1978). Topics in group rings. New York, US: Marcel Dekker.
There are 14 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Turgut Hanoymak 0000-0002-3822-2202

Ömer Küsmüş 0000-0001-7397-0735

Early Pub Date April 29, 2023
Publication Date April 30, 2023
Submission Date April 2, 2022
Published in Issue Year 2023 Volume: 28 Issue: 1

Cite

APA Hanoymak, T., & Küsmüş, Ö. (2023). A Generalization of G-Nilpotent Units in Commutative Group Rings to Direct Product Groups. Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 28(1), 8-18. https://doi.org/10.53433/yyufbed.1097581

Cited By

On the Unit Group of the Integral Group Ring Z(S_3×C_3)
Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi
https://doi.org/10.53433/yyufbed.1361776