Jacobson’s Lemma for modules
Ricerche di Matematica, 2026 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Basım Tarihi: 2026
- Doi Numarası: 10.1007/s11587-026-01195-5
- Dergi Adı: Ricerche di Matematica
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH, Materials Science & Engineering Collection (ProQuest), Technology Collection (ProQuest)
- Anahtar Kelimeler: (b, (r, Generalized inverse, Jacobson’s Lemma, Module decomposition, Ring decomposition, c)-inverse, f)-inverse
- Ankara Üniversitesi Adresli: Evet
Özet
Let R be a ring with identity and M a right R-module. Define S=EndR(M) as the ring of all right R-module endomorphisms of M. Motivated by the (b, c)-inverse in ring theory, an element m∈M is called (r,f)-invertible if there exists φ∈M∗=HomR(M,R) such that φ∈rRφ∩φSf, φmr=r and fmφ=f where r∈R and f∈S. In this paper, we present Jacobson’s Lemma for modules, and use it to obtain relations between the (r, f)-invertibility of elements in modules. As an application, we establish direct-sum decompositions for the module M, its endomorphism ring S and the base ring R, using (r, f)-inverses. These results extend and unify known facts from the theory of (b, c)-inverses in rings to the module setting.