On GCO-modules and M-small modules

Abstract

Let M be a right R-module. Define $Z^{ast}(N)$ ($delta^{ast}_M (N)$ to be the set of elements n $in$ N for any R-module N in $sigma [M]$ such that nR is an M-small (respectively $delta$-M-small) module. In this note it is proved that M is a GCO-module if and only if every M-small module in $sigma[M]$ is M-projective if and only if every $delta$-[M]-small module in $sigma[M]$ is M-projective. Also, if $M/delta_ M$ (M) is semisimple then M is a GCO-module if and only if M is an SI-module.