A generalization of the generalized Drazin inverse in rings

Tien N. Q., KOŞAN M. T., Quynh T. C., ÖZDİN T.

Journal of Algebra and its Applications, 2025 (SCI-Expanded) identifier

  • Yayın Türü: Makale / Tam Makale
  • Basım Tarihi: 2025
  • Doi Numarası: 10.1142/s0219498827500071
  • Dergi Adı: Journal of Algebra and its Applications
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Anahtar Kelimeler: Drazin inverse, EP-element, generalized Drazin inverse, Jacobson radical, nilpotent element, Quasi-nilpotent element, quasi-polar element, spectral idempotent
  • Erzincan Binali Yıldırım Üniversitesi Adresli: Evet

Özet

The notions of (quasi-) polarity and (generalized) Drazin invertibility of an element a in a ring R, defined upon the set of (quasi-) nilpotent elements, are extended by using the set ∇(R) = {a ∈ R : 1 − au is unit in R for all unit u ∈ R with ua = au}. This set is adapted from the set ∆(R), the largest Jacobson radical subring of R closed by multiplication by units. Uniqueness of the spectral idempotent usually associated to (quasi-) polar decompositions remains, and can be defined by minimal properties. This leads to the definition of a unique Drazin-type inverse for a wider collection of decompositions, encompassing the quasi-polar decompositions.