Hereditarily and cohereditarily projective modules

TL;DR: In this article, it was shown that a ring R will be right-hereditary if all projective right R -modules are hereditarily projective, i.e., the image of any homomorphism of P into a free module of finite rank is again projective.

Abstract: This chapter describes the hereditarily and cohereditarily projective modules. There is a similarity between the study of n-firs and that of right semihereditary rings. In each case, one works with projective right modules P over the given ring R with the property that the image of any homomorphism of P into a free module of finite rank is again projective. For R semihereditary, all finitely generated projective right R -modules have this property. It is found that If P is a finitely generated projective module over a ring R , then P * will again be projective, and finitely generated, and P ** is naturally isomorphic to P ; it is well known that * gives an anti-isomorphism between the categories of finitely generated projective right and left R -modules. It is observed that a ring R will be right-hereditary if all projective right R -modules are hereditarily projective.