Rings in which elements are the sum of a unit and a non-unit

HALICIOĞLU S., ÜNGÖR B.

Bulletin of the Belgian Mathematical Society - Simon Stevin, cilt.31, sa.4, ss.501-515, 2024 (SCI-Expanded) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 31 Sayı: 4
  • Basım Tarihi: 2024
  • Doi Numarası: 10.36045/j.bbms.240318a
  • Dergi Adı: Bulletin of the Belgian Mathematical Society - Simon Stevin
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.501-515
  • Anahtar Kelimeler: fine ring, semiprimitive ring, semisimple ring, special clean ring, Unit element, unit-fusible ring
  • Ankara Üniversitesi Adresli: Evet

Özet

We introduce a class of rings, called Harman rings, which is a proper subclass of semiprimitive rings. Let R be a ring with identity. Then R is called Harman if every non-zero element is the sum of a unit and a non-unit in R. We investigate relations between Harman rings and some important classes of rings, such as semisimple rings, fine rings, unit-fusible rings, special clean (equivalently, unit-regular) rings and Boolean rings. In particular, we prove that R is a local Harman ring if and only if R is a division ring. A notable result states that for every positive integer n, the matrix ring Mn(R) is Harman if and only if R is Harman. This implies, in particular, that any non-zero square matrix over a division ring can be expressed as the sum of a unit matrix and a non-unit matrix.