Commutative rings in which every finitely generated ideal is quasi-projective

• Published in: Journal of pure and applied algebra Vol. 215; no. 10; pp. 2504 - 2511
• Main Authors: Abuhlail, J., Jarrar, M., Kabbaj, S.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.10.2011
• Subjects: 13C13; 16D90; 13F05; 16B50; 16D40; 13B05
• ISSN: 0022-4049; 1873-1376
• DOI: 10.1016/j.jpaa.2011.02.008

This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. We provide some preliminaries on quasi-projective modules over commutative rings. Then we investigate the correlation with the well-known Prüfer conditions; that is, we prove that this class of rings stands strictly between the two classes of arithmetical rings and Gaussian rings. Thereby, we generalize Osofsky’s theorem on the weak global dimension of arithmetical rings and partially resolve Bazzoni–Glaz’s related conjecture on Gaussian rings. We also establish an analogue of Bazzoni–Glaz results on the transfer of Prüfer conditions between a ring and its total ring of quotients. We then examine various contexts of trivial ring extensions in order to build new and original examples of rings where all finitely generated ideals are subject to quasi-projectivity, marking their distinction from related classes of Prüfer rings.