亚投射模与亚内射模

SUBPROJECTIVE MODULES AND SUBINJECTIVE MODULES

通过引进模的内射根，给出亚内射模的定义．用亚投射模和亚内射模，给出任意环Ｒ中非０单投射和非０单内射模的存在性的等价该划．同时我们考虑了亚投射模和亚内射模的一些性质，讨论了亚内射模和亚投射模的结构．最后我们举出例子说明：亚投射极未必是投射模，亚内射模未必是内射模，并且投射模也未必是亚投射模。

in this paper, we have introduced two concepts: injective radical andsubinjective module, and established some elementary relations between subinjective,semi-simple injective and simple injectve modules. We have also established some relations between subprojective,semi-simple projective and simple projective modules. Let R be an associative ring with identity. Then it follows that there exists a non-zero subprojective module over R if and only if there exists a non-zero simple projecticve module over R .In the same way, it follows that there exists a non-zero subinjective module over R if and only if there exists a non-zero subinjective module over R if and only if there exists a non-zero simple injective module over R.Using the short exact sequences, we have considered some properties of the subprojective modules and the subinjective modules. We have investigated the structures of the subprojective modules and the subinjective modules with the finite cogeneration.In the examples, it shows that there exists a subprojective module which is not a projective module.and there exists a projective module which is not a subprojective module. In example 3, it follows that it is not necessarily true that every subinjective module is an injective module. In fact, there also exists all injective module which is not a subinjective module.