Finitely Generated Flat Modules and a Characterization of Semiperfect Rings

Abstract

For a ring S, let K ₀(FGFl(S)) and K ₀(FGPr(S)) denote the Grothendieck groups of the category of all finitely generated flat S-modules and the category of all finitely generated projective S-modules respectively. We prove that a semilocal ring Ris semiperfect if and only if the group homomorphism K ₀(FGFl(R)) → K ₀(FGFl(R/J(R))) is an epimorphism and K ₀(FGFl(R)) = K ₀(FGPr(R)).

Dedicated to the memory of Professor Ahmad Shamsuddin.

Key Words:

• Flat modules
• Semilocal rings
• Lifting idempotents
• Semiperfect rings

6. Acknowledgments

The first author was supported by Gruppo Nazionale Strutture Algebriche e Geometriche e loro Applicazioni of Istituto Nazionale di Alta Matematica (Italy) and Ministero dell'Istruzione, dell'Università e della Ricerca (progetto di ricerca di rilevante interesse nazionale “Nuove prospettive nella teoria degli anelli, dei moduli e dei gruppi abeliani”), Italy. The research of the second author was partially supported by the DGESIC (Spain) through the project PB98-0873, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. The work of the third author was partially supported by RFFR 99-01-00469 (Russia) and Consiglio Nazionale delle Ricerche (Italy) through the International Short-Term Mobility Program 2000.

Notes

Dedicated to the memory of Professor Ahmad Shamsuddin.