For a torsion radical F (arising
from an idempotent filter of right ideals) and a unital right R-module M over a ring
R, let DM be the F-divisible hull DM∕M = F(EM∕M), where EM is the injective
hull of M.
Let 0≠β : N → M be any nonzero homomorphism whatever from any F-dense
submodule N ⊆ M. Then M is F-quasi-injective if each such β extends to a
homomorphism of M → M; M is F-monic if β is monic; M is F-co-monic if βN ⊆ M
is F-dense.
Each module M has a natural F-quasi-injective envelope JM inside
M ⊆ JM ⊆ DM.
Theorem III. Form the R-endomorphism rings Δ = EndJM and Λ = EndDM, and
Λ# = {λ ∈ Λ∣λM = 0}⊆ Λ, the annihilator subring of M.
When M is F-monic and F-co-monic and FM = 0, then
(1) Λ# is exactly the annihilator Λ# = {λ ∈ Λ∣λJM = 0} of the submodule
JM ⊆ DM and Λ# ⊆ Λ is an ideal;
(2) Δ≅Λ∕Λ#;
(3) Δ is a division ring.
For a torsion radical F and a torsion preradical G, let IM be the (F,G)-injective
hull of M; and, more generally, Λ the ring of all those R-endomorphisms of IM with
G-dense kernels. The above is derived as a special case where G = 1 is the identity
functor and IM = DM.
Theorem II.
- M is (F,G)-quasi-injective ⇔ ΛM ⊆ M.
- The (F,G)-quasi-infective hull JM of M exists and JM = M + ΛM.
- JM is the unique smallest (F,G)-quasi-injective module with M ⊆ JM ⊆
IM.
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