$(n,d)$-COCOHERENT RINGS, $(n,d)$-COSEMIHEREDITARY RINGS AND $(n,d)$-$V$ -RINGS

Abstract

. Let R be a ring, n be an non-negative integer and d be a positive integer or ∞ . A right R -module M is called ( n,d ) ∗ -projective if Ext 1 R ( M,C ) = 0 for every n -copresented right R -module C of injective dimension ≤ d ; a ring R is called right ( n,d ) -cocoherent if every n -copresented right R -module C with id ( C ) ≤ d is ( n +1)-copresented; a ring R is called right ( n,d ) -cosemihereditary if whenever 0 → C → E → A → 0 is exact, where C is n -copresented with id ( C ) ≤ d , E is finitely cogenerated injective, then A is injective; a ring R is called right ( n,d ) - V -ring if every n -copresented right R -module C with id ( C ) ≤ d is injective. Some characterizations of ( n,d ) ∗ -projective modules are given, right ( n,d )-cocoherent rings, right ( n,d )-cosemihereditary rings and right ( n,d )- V -rings are characterized by ( n,d ) ∗ -projective right R -modules. ( n,d ) ∗ -projective dimensions of modules over right ( n,d )-cocoherent rings are investigated.