Essential and Retractable Galois Connections

Abstract

Abstract. For bounded lattices, we introduce certain Galois connections, called (cyclically) essential, retractable and UC Galois connections, which behave well with respect to concepts of module-theoretic nature involving essentiality. We show that essential retractable Galois connections preserve uniform dimension, whereas essential retractable UC Galois connections induce a bijective correspondence between sets of closed elements. Our results are applied to suitable Galois connections between submodule lattices. Cyclically essential Galois connections unify semi-projective and semi-injective modules, while retractable Galois connections unify retractable and coretractable modules. IntroductionIf M and N are two right R-modules, then it is well-known that Hom R (M, N ) has a structure of (End R (N ), End R (M ))-bimodule. An interesting classical problem in module theory is to relate properties of the modules M and N with properties of the bimodule Hom R (M, N ), and in particular, to relate properties of a module with properties of its endomorphism ring. The present paper is motivated by this general problem, and its goal is to show how one can efficiently use Galois connections in order to obtain and clarify the desired results by using easier to handle and more transparent conditions.We introduce certain Galois connections, called (cyclically) essential, retractable and UC Galois connections, between two bounded lattices. Essential retractable Galois connections turn out to preserve uniform dimension. Also, we prove that essential retractable UC Galois connections between two bounded modular lattices A and B induce a bijective correspondence between the set of closed elements of A and the set of closed elements of B. As an application, we show how the extending property transfers through essential retractable UC Galois connections. Our paper expands the approach from [4], not only dualizing, but also generalizing results involving essentiality in bounded (modular) lattices. They can be applied to some particular Galois connections between submodule lattices, previously pointed out by Albu and Nȃstȃsescu [1,. Then retractable Galois connections particularize to retractability and coretractability of modules, cyclically essential Galois connections particularize to semi-projectivity and semi-injectivity of modules, and our results immediately yield and unify recent properties from [2,10]. Thus it becomes clear that both retractability and coretractability are instances of the same notion, and this is also the case for semi-projectivity and semi-injectivity. When M is a right R-module and N is an M -faithful right R-module, we deduce a module-theoretic result of Zelmanowitz on the existence of a bijective correspondence between the sets of closed right R-submodules of M and closed right End R (M )-submodules of Hom R (M, N ) [13, Theorem 1.2]. Special Galois connectionsLet us recall the concept of (monotone) Galois connection (e.g., see [8]).Definition 2.1. Let (A, ≤) and (B, ≤) be lattices. A Galois connecti...

Rickart and Dual Rickart Objects in Abelian Categories

“…In order to generalize these results we use Galois connections between subobject lattices in abelian categories and techniques for special Galois connections developed in [6,7,27].…”

Rickart and Dual Rickart Objects in Abelian Categories: Transfer via Functors

“…Secondly, our concepts generalize to the level of abelian categories Rickart and dual Rickart modules in the sense of Lee, Rizvi and Roman [18,19,20], and in particular, Baer and dual Baer modules studied by Rizvi and Roman [25,26] and Keskin Tütüncü, Smith, Toksoy and Tribak [16,17]. A unified approach of Baer and dual Baer modules via Baer-Galois connections was given by Olteanu in [24], following the approach by Crivei from [5]. The root of (dual) Baer and (dual) Rickart modules traces back to the work of Kaplansky [15] on Baer rings and Maeda [22] on Rickart rings.…”