We introduce a generalization of the Gorenstein injective modules:the Gorenstein FPn-injective modules(denoted by GI_(n)).They are the cycles of the exact complexes of injective modules that remain exact when we apply...We introduce a generalization of the Gorenstein injective modules:the Gorenstein FPn-injective modules(denoted by GI_(n)).They are the cycles of the exact complexes of injective modules that remain exact when we apply a functor Hom(A,-),with A any FP_(n)-injective module.Thus,GL_(o)is the class of classical Gorenstein injective modules,and GI_(1)is the class of Ding injective modules.We prove that over any ring R,for any n≥2,the class GI_(n)is the right half of a perfect cotorsion pair,and therefore it is an enveloping class.For n=1 we show that GI_(1)(i.e.,the Ding injectives)forms the right half of a hereditary cotorsion pair.If moreover the ring R is coherent,then the Ding injective modules form an enveloping class.We also define the dual notion,that of Gorenstein FP_(n)-projectives(denoted by GP_(n)).They generalize the Ding projective modules,and so,the Gorenstein projective modules.We prove that for any n≥2 the class GP_(n)is the left half of a complete hereditary cotorsion pair,and therefore it is special precovering.展开更多
文摘We introduce a generalization of the Gorenstein injective modules:the Gorenstein FPn-injective modules(denoted by GI_(n)).They are the cycles of the exact complexes of injective modules that remain exact when we apply a functor Hom(A,-),with A any FP_(n)-injective module.Thus,GL_(o)is the class of classical Gorenstein injective modules,and GI_(1)is the class of Ding injective modules.We prove that over any ring R,for any n≥2,the class GI_(n)is the right half of a perfect cotorsion pair,and therefore it is an enveloping class.For n=1 we show that GI_(1)(i.e.,the Ding injectives)forms the right half of a hereditary cotorsion pair.If moreover the ring R is coherent,then the Ding injective modules form an enveloping class.We also define the dual notion,that of Gorenstein FP_(n)-projectives(denoted by GP_(n)).They generalize the Ding projective modules,and so,the Gorenstein projective modules.We prove that for any n≥2 the class GP_(n)is the left half of a complete hereditary cotorsion pair,and therefore it is special precovering.
