Good dualities and strongly quasi-injective modules

Menini, Claudia; Orsatti, Adalberto

doi:10.1007/bf01811724

Cited by 23 publications

(8 citation statements)

References 17 publications

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“…Hence 2 is equal to It* iff 2 is finer than ~t*. (2) so that #* is, in this case, strictly finer than #.…”

Section: ) the Assignement H-~h• Gives A Bijection Between The Set Omentioning

confidence: 92%

“…Thus, by using Baer's criterion for injectivity, one gets that RK is an injective object for ~'0,,' As ~r is equivalent to Qr, ~'~,, and ~'0K have the same simple objects and hence RK is an injective cogenerator of ~'~,,. Following result is Lemma 9.2 of [2]. We recall it here for benefit of the reader as we are going to apply it.…”

Section: Proposition In the Notations Of 8 ~: Is The Filter Of All mentioning

confidence: 94%

See 1 more Smart Citation

A characterization of linearly compact modules

Menini

1985

Math. Ann.

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“…Hence 2 is equal to It* iff 2 is finer than ~t*. (2) so that #* is, in this case, strictly finer than #.…”

Section: ) the Assignement H-~h• Gives A Bijection Between The Set Omentioning

confidence: 92%

Section: Proposition In the Notations Of 8 ~: Is The Filter Of All mentioning

confidence: 94%

A characterization of linearly compact modules

Menini

1985

Math. Ann.

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“…In this paper, we apply results of Menini and Orsatti [9] and Zelmanowitz [20] developed further in this direction to study the bicommutator B = R * * of the module R K, when R K is a minimal (injective) cogenerator and K T is quasi-injective. The bicommutator may also be viewed as the K-adic completion [1] of the ring R. Now clearly R ⊗ R R * = R ⊗ R K = K ∈ σ [ R K], so if the R K-topology on R R is of finite type, then Theorem A.3 implies that B R is the pure-injective envelope of R R .…”

Section: Theorem B (Theorem 5 Proposition 8) Let R K T Be An R-t-bimmentioning

confidence: 98%

“…While it may seem standard to prove such results only when R K is the minimal injective cogenerator, we follow Menini and Orsatti [9] by noting that a parallel theory may be developed under the weaker assumption that R K is the minimal cogenerator.…”

mentioning

confidence: 98%

Applications of Duality to the Pure-injective Envelope

Herzog

2006

Algebr Represent Theor

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Given an R-T-bimodule R K T and an R-S-bimodule R M S , we study how properties of R K T affect the K-double dual M * * = Hom T [Hom R (M, K ), K], considered as a right S-module. If R K is a cogenerator, then for every R-S-bimodule, the natural morphism M : M → M * * is a pure-monomorphism of right S-modules. If R K is the minimal (injective) cogenerator and K T is quasi-injective, then M * * is a pure-injective right S-module. If R K is the minimal (injective) cogenerator, and T = End R K, it is shown that K T is quasi-injective if and only if the K-topology on R is linearly compact. If the R K-topology on R is of finite type, then the natural morphism R : R → R * * is the pure-injective envelope of R R as a right module over itself.Key words bicommutator · M-topology · minimal injective cogenerator · pure-injective envelope · quasi-injective Mathematics Subject Classifications (2000) 16D50 · 16D90 · 16S90Let R and T be associative rings with identity, and R K T an R-T-bimodule. The bimodule R K T induces an additive endofunctor (−) * * : R-Mod → R-Mod, the Kdouble dual, of the category of left R-modules. It is defined byThere is a natural transformation : 1 R−Mod → (−) * * from the identity functor to the K-double dual, given by the evaluation morphism M : M → M * * , M (m) : ξ → ξ(m). The functorial nature of (−) * * implies that if M has an R-S-bimodule

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“…Over the years, several authors 16,19,22,25 , for example investigated a dual notion of quasi-progenerators, called quasi-duality modules in w x w x 14, 15 , and cotilting modules 2, 4, 9 , dual to tilting modules, have been a central topic of recent investigations in module theory. Both types of modules induce generalizations of Morita duality.…”

Section: S R Rmentioning

confidence: 99%