相对Gorenstein投射模

摘要

Let A be an extension ring of a ring B,that is,B is a subring of A with the same identity.We denote by P(A,B) the category of all the relatively projective modules.For this extension B → A,we introduce relatively Gorenstein-projective modules.As Gorenstein-projective modules are closely related to projective modules and there are some good results about Gorenstein dimensions,we want to give a similar relationship between relatively Gorenstein-projective modules and relatively projective modules.The main results are:(1) Let B → A be an extension of rings with the same identity.Then the category of all the relatively Gorenstein projective modules is relatively resolving.(2) Let B → A be an extension of rings with the same identity.If gl.dim(A,B) ≤ n,then every relatively Gorenstein-projective module is relatively projective,where gl.dim(A,B) represents the supreme of relatively projective dimension of all the A-modules.%设环A是环B的扩张环,即B是与A有相同单位的A的子环.记P(A,B)是由所有相对投射模构成的范畴.对于扩张B → A,本文介绍相对Gorenstein投射模的概念.由于Gorenstein投射模与投射模具有紧密的联系,并且关于Gorenstein维数有较好的性质,本文想给出相对Gorenstein投射模和相对投射模之间类似的关系.本文主要结果是:(1)设B →A是具有相同单位的环的扩张,则由所有相对Gorenstein投射模构成的范畴是相对可解的.(2)设B → A是具有相同单位的环的扩张,若gl.dim(A,B)≤n,则每一个相对Gorenstein投射模都是相对投射的,其中gl.dim(A,B)表示所有A-模的相对投射维数的上确界.