Weakly distributive modules. Applications to supplement submodules

Abstract

Abstract. In this paper, we define and study weakly distributive modules as a proper generalization of distributive modules. We prove that, weakly distributive supplemented modules are amply supplemented. In a weakly distributive supplemented module every submodule has a unique coclosure. This generalizes a result of Ganesan and Vanaja. We prove that π -projective duo modules, in particular commutative rings, are weakly distributive. Using this result we obtain that in a commutative ring supplements are unique. This generalizes a result of Camillo and Lima. We also prove that any weakly distributive ⊕-supplemented module is quasi-discrete.

On the Notion of Strong Irreducibility and Its Dual

“…Modules such that all submodules have unique supplements were studied by Ganesan and Vanaja [10]. Weakly distributive modules do have this property (see [8] From Corollary 1.9 we get the following statement. The following property should be compared to Stephenson's characterizations of distributive modules which says that a module M is distributive if and only if Hom(P/(P ∩ Q), Q/(P ∩ Q)) = 0 for any submodules P, Q of M (see [27]).…”

“…Modules such that all submodules have unique supplements were studied by Ganesan and Vanaja [10]. Weakly distributive modules do have this property (see [8]). A submodule U of a module M is said to be weakly distributive if U = (U ∩ X) + (U ∩ Y ) for any submodules X, Y with X + Y = M. Equivalently U is a weakly +-distributive element in the dual lattice L • of the lattice L = (L(M), ∩, +, 0, M).…”

Principally ⨁-G-〖Rad〗_g-supplemented modules

“…We said that a submodule 𝐴 of a module 𝑀 is weak distributive if 𝐴 = (𝐴 ∩ 𝑋) + (𝐴 ∩ 𝑌) for all submodules 𝑋, 𝑌 of 𝑀 with 𝑋 + 𝑌 = 𝑀. A module 𝑀 is said to be weakly distributive if every submodule of 𝑀 is a weak distributive submodule of 𝑀 [2]. A module 𝑀 is said to have the summand sum property (SSP) if the sum of any two direct summands of 𝑀 is again a direct summand of 𝑀 [1].…”

ss-lifting modules and rings

“…Recall from [1] that a submodule U is called a weak distributive of M if U = (U ∩ X) + (U ∩Y ) for all submodules X,Y ≤ M such that M = X +Y . A module M is said to be weakly distributive if every submodule of M is a weak distributive submodule of M.…”