MODULES WITH FI-EXTENDING HULLS

It is shown that every finitely generated projective module PR over a semiprime ring R has the smallest FI-extending essential module extension $H_{\frak{FI}}(P_R)$ (called the absolute FI-extending hull of PR) in a fixed injective hull of PR. This module hull is explicitly described. It is proved that $\widehat{Q}_{\frak{FI}}({\rm End}(P_R))\cong {\rm End}(H_{\frak{FI}}(P_R))$, where $\widehat{Q}_{\frak{FI}}({\rm End}(P_R))$ is the smallest right FI-extending right ring of quotients of End(PR) (in a fixed maximal right ring of quotients of End(PR). Moreover, we show that a finitely generated projective module PR over a semiprime ring R is FI-extending if and only if it is a quasi-Baer module and if and only if End(PR) is a quasi-Baer ring. An application of this result to C*-algebras is considered. Various examples which illustrate and delimit the results of this paper are provided.