2011/08/10 by Yongduo Wang, Wang, Yongduo
Mathematics · #16D10 #16D50 #Advanced Topics in Algebra #Commutative Algebra and Its Applications #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.RA #msc:16D10 #msc:16D50
paper · pdf · doi:10.48550/arxiv.1108.2083
21
arxiv created 2011/08/10 · openalex publication_date 2011/08/10 · arxiv updated 2011/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be a left module over a ring R and I an ideal of R. We call (P, f) a (locally)projective I-cover of M if f is an epimorphism from P to M, P is (locally)projective, Kerf⊆ IP, and whenever P=Kerf+X, then there is a projective summand Y of P in Kerf such that P=Y⊕ X. This definition generalizes (locally)projective covers. We characterize I-semiregular and I-semiperfect rings which are defined by Yousif and Zhou [19] using (locally)projective I-covers in section 2 and 3. I-semiregular and I-semiperfect rings are characterized by projectivity classes in section 4. Finally, the notion of I-supplemented modules are introduced and I-semiregular and I-semiperfect rings are characterized by I-supplemented modules. Some well known results are obtained as corollaries.