Group rings that are UJ rings

KOŞAN M. T., Zemlicka J.

COMMUNICATIONS IN ALGEBRA, cilt.49, sa.6, ss.2370-2377, 2021 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 49 Sayı: 6
  • Basım Tarihi: 2021
  • Doi Numarası: 10.1080/00927872.2020.1871000
  • Dergi Adı: COMMUNICATIONS IN ALGEBRA
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Sayfa Sayıları: ss.2370-2377
  • Anahtar Kelimeler: Commutator subgroup, group ring, Jacobson radical, locally finite 2-group, solvable group, trivial Morita context, UJ-rings, unit
  • Gazi Üniversitesi Adresli: Evet

Özet

The set Delta(R) of all elements r of a ring R such that 1 + ru is a unit for every unit u extends the Jacobson radical J(R). R is a UJ ring (Delta U ring, respectively) if its units are of the form 1 +J(R) (1 + Delta(R), respectively). Using a local characterization of Delta U rings, we describe structure of group rings that are UJ rings; if RG is a UJ group ring, then R is a UJ ring, G is a 2-group and, for every nontrivial finitely generated subgroup H of G, the commutator subgroup of H is proper subgroup of H. Conversely, if R is a W ring and G a locally finite 2-group, then RG is a UJ ring. In particular, if G is solvable, RG is a UJ ring if and only if R is UJ and G is a 2-group.