强素子模

摘要

讨论了环与模范畴中一个重要的子模类——强素子模的一些性质.类似于素子模研究中的一些特有性质,给出了强素子模的一些等价条件,即强素子模与其商模和商模的零子模之间的关系,证明了若N是M的子模,L是M的强素子模,使得N(￠)L,则(L ∩ N∶N)=(L∶M)且L∩N是N的强素子模；通过素理想与极大理想等价的一些条件给出了素子模与强素子模的等价条件；讨论了强素子模与一种特殊模类——完全忠实模的关系.%The strong prime submodules and isolated submodules play important roles in the study of rings and categories of modules. In this paper, some related properties of strong prime submodules are discussed. We will give some equivalent conditions which are similar to some specific properties of prime submodules, that is,some relations among strong submodules, factor modules and the zero submodules. It is proved that if N is a submodule of M,and L is a strong prime submodule of M,such that N￠L. Then (L∩ N :N) = (L : M) and L∩ N is a strong prime submodule of N. Some equivalent conditions for strong prime submodules and prime submodules are given by some conditions of equivalence of prime ideal and maximal ideal. The relation between strong prime submodules and the special module class, that is, fully faithful modules is also discussed.