Small projective modules (Q1057352)

The paper is devoted to the investigation of small projective modules which are a generalization of projective modules by using small submodules in a sense analogous to pure projective modules being a generalization of projective ones by using pure submodules. A submodule K of a module M is called small if \(K+L=M\) implies \(L=M\) for every submodule L of M. An R-module M is hollow if every proper submodule of M is small. An R-module is small projective if for any epimorphism \(g: B\to A\) whose kernel is small, \(g\circ Hom_ R(M,B)=Hom_ R(M,A).\) In the paper some properties of small projective modules are obtained. The main result is Theorem 1.15. Let M be a small projective hollow module and S be the endomorphism ring of M. Then (i) \(J(S)=\{\alpha \in S|\) Im \(\alpha\) is a small in \(M\}\) ; (ii) S/J is a von Neumann regular ring; (iii) J(M) is small in M iff \(Hom_ R(M,J(M))=J(S)\). (Here J(M) is the Jacobson radical of M.)