A Survey on Projective Modules

An R-module M is called n-projective (n a positive integer) if the functor Extn^-1R (M,−) is a right exact functor. We say that an R-module M is (n + ½ )-projective if it is the cokernel of an *n-projective preenvelope. The characterization of the class of (n + ½ )-projective R-modules is explained in this paper. At first we prove some properties of n-projective modules, and investigate the relation between n-projective and (n+ ½ )-projective modules. Then we study when the exactness of B - C - 0 implies the exactness of Extn^-1R (M,B) _ Extn^-1_R (M,C)_ 0 for every (n + ½ )-projective module M.