Vol. 66, No. 1, 1976

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Generalized monoform and quasi injective modules

John Dauns

Vol. 66 (1976), No. 1, 49–65

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 0β : 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 ΛλM = 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 ΛλJM = 0} of the submodule JM DM and Λ# Λ is an ideal;

(2) ΔΛΛ#;

(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
Received: 25 July 1974
Published: 1 September 1976
John Dauns