Generalized monoform and quasi injective modules

John Dauns

Vol. 66 (1976), No. 1, 49–65

DOI: 10.2140/pjm.1976.66.49

Abstract

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.

1. M is (F,G)-quasi-injective ⇔ ΛM ⊆ M.
2. The (F,G)-quasi-infective hull JM of M exists and JM = M + ΛM.
3. JM is the unique smallest (F,G)-quasi-injective module with M ⊆ JM ⊆ IM.

Mathematical Subject Classification 2000

Primary: 16A52, 16A52

Secondary: 18E40

Milestones

Received: 25 July 1974

Published: 1 September 1976