Relative copure injective and copure flat modules

Abstract

Let R be a ring, n a fixed nonnegative integer and I n ( F n ) the class of all left (right) R -modules of injective (flat) dimension at most n . A left R -module M (resp., right R -module F ) is called n -copure injective (resp., n -copure flat) if Ext 1 ( N , M ) = 0 (resp., Tor 1 ( F , N ) = 0 ) for any N ∈ I n . It is shown that a left R -module M over any ring R is n -copure injective if and only if M is a kernel of an I n -precover f : A → B of a left R -module B with A injective. For a left coherent ring R , it is proven that every right R -module has an F n -preenvelope, and a finitely presented right R -module M is n -copure flat if and only if M is a cokernel of an F n -preenvelope K → F of a right R -module K with F flat. These classes of modules are also used to construct cotorsion theories and to characterize the global dimension of a ring under suitable conditions.