强τ-Extending模

Stronglyτ-Extending Modules

设τ=(T,F)是一个遗传挠理论.首先引入了强τ-Extending模的概念,研究了强τ-Extending模的性质,讨论了τ-Extending模与强τ-Extending模之间的关系,举例说明了τ-Extending模未必是强τ-Extending模.然后研究了强τ-Extending模关于直和因子以及直和的封闭性,结果表明:强τ-Extending模关于直和因子封闭,但它关于直和不封闭.进而,给出了强τ-Extending模关于直和封闭的局部条件,证明了:τ-挠自由的强τ-Extending模与τ-挠模的直和仍是强τ-Extending模.最后,对于M=i∈IMi(|I|≥2),给出了M是强τ-Extending模的等价刻画.

Letτ=(T,F)be a hereditary torsion theory.The concept of stronglyτ-Extending module is introduced,several properties of stronglyτ-Extending module are studied,and the relations betweenτ-Extending module and stronglyτ-Extending module are discussed,an example is given to show thatτ-Extending module is not necessarily stronglyτ-Extending.The closure of strongτ-Extending module under direct summand and direct sum are considered.The results show that strongτ-Extending module is closed with respect to the direct summand,but it is not closed with respect to the direct sum.Moreover,the local conditions that direct sum of strongτ-Extending module is also strongτ-Extending are given,it is proved that the direct sum ofτ-torsion free stronglyτ-Extending module andτ-torsion module is also stronglyτ-Extending.Finally,some equivalent characterizations that M is stronglyτ-Extending module are given for M=i∈IMi(|I|≥2).