t-UC模的若干等价刻画

Several Equivalent Characterizations of t-UC Modules

设τ是遗传挠理论,基于Asgari和Ceken等人引入的t-本质子模和τ-UC模的概念,利用环模理论的研究方法,给出了t-UC模的概念。若M的任意子模在M中都存在唯一的t-闭包,则称M是t-UC模。通过举例说明了t-UC模和UC模之间的关系,讨论了t-UC模的若干等价刻画,并给出了两条推论:(1)若M是t-UC模,则对任意N≤tcM,NM是t-UC模和UC模;(2)当且仅当M的任意子模是t-UC模时,M是t-UC模。进而,当M=⊕i∈IMi是t-UC模时,证明了M是t-extending模当且仅当对任意i∈I,M是t-extending模,且对满足K∩Mi■Z2(M)(i∈I)产M的任意t-闭子模K,K是M的直和因子。

Let τ be a hereditary torsion theory,based on the concept of t-essential submodules and τ-UC modules introduced by Asgari and Ceken et al,using methods of rings and modules theory,the concept of t-UC module is given. An module M is called a t-UC module provided every submodule has a unique t-closure. An example is given to illustrate the relationship between t-UC modules and UC modules,some equivalent characterizations of t-UC modules are discussed,and the following two points are proved:(1) If M is a t-UC module then N/M is a t-UC module and a UCmodule for every N ≤tcM;(2) M is a t-UC module if and only if every submodule of M is a t-UC module. Moreover,when M = ⊕i∈IMi is a t-UC module,it is proved that M is t-extending if and only if the module Miis t-extending for each i ∈ I and every t-closed submodule K of M with K ∩ Mi■Z2(M)( i ∈ I) is a direct summand of M.