Locally injective torsion modules (Q1075380)

Let R be a commutative ring and F a Gabriel topology on R. In the paper: ''Torsion theories over commutative rings'', J. Algebra 101, 136-150 (1986), the author and \textit{E. Barbut} showed that the ring R is F-local if for any F-torsion module T there is an isomorphism between T and the direct sum of the family \(\{T_ M:\quad M\quad a\quad \max imal\quad ideal\quad of\quad R\}.\) The paper discusses those rings R which satisfy the property that for all F-torsion modules T, T is F-injective if and only if F is locally F-injective, however two different possible definitions of locally F-injective are preposed, in one the above condition characterizes the F-local rings, for the other definition which is more natural but more difficult to characterize, every F-local ring has the property but not conversely.