Decomposing torsion modules

Rings that are FGC relative to filters of ideals

“…By [5,Theorem 1], if J* is a filter of ideals of R, then R is an ^-local ring if and only if =0M 6mS pec« T(M) for all J^-pretorsion modules T if and only if r = 0 M e m s p e c i { T(M) for all cyclic #"-pretorsion modules T. There are a number of results relating to ^-local that have been proved for Gabriel topologies, which are in fact true for filters of ideals. Those that we shall be needing are [ The reader might suspect that if J 5 " is a filter of ideals of R, then R is an J^-maximal ring if and only if R is an J^-almost maximal ring.…”

“…Our definition of #"-local is inconsistent with the previously defined 3F-& property. We allow this inconsistency because of our earlier use of J*-local as a generalization of /i-local in [3], [4] and [5]. If T is an R-module and MemspecR, then use the notation…”

“…Also R will always denote a ring. J 5 " is a filter of ideals of R if J* is a nonempty set of ideals of R satisfying: if / e i * and J is an ideal of R with laJ, then JeJ^, and if /, ./eJ* then In JeJ*. A Gabriel topology of R is a filter of ideals !F of R satisfying: if J e !F and / is an ideal of R with (l:x)e& r for all xeJ, then Ie^.…”

“…We allow this inconsistency because of our earlier use of J*-local as a generalization of /i-local in [3], [4] and [5]. If T is an R-module and MemspecR, then use the notation The following is comparable to [2, Theorem 2.9], which is stated for integral domains with the classical torsion theory.…”

“…Since R is an J*-local ring, |mspec(/)|<oo, and by[5, Theorem 1] and [3, Theorem 1], R/I^@ MemspccR RJI M . If MemspecR-mspec(/), then RJI M^{ 0}.…”