Ding injective envelopes in the category of complexes

Abstract

A complex $X$ is called Ding injective if there exists an exact sequence of injective complexes $\dots \to E_1\to E_0\to E_{-1}\to \dots$ such that $X=Ker(E_0\to E_{-1})$, and the sequence remains exact when the functor $Hom(A,-)$ is applied to it, for any $FP$-injective complex $A$. We prove that, over any ring $R$, a complex is Ding injective if and only if it is a complex of Ding injective modules. We use this to show that the class of Ding injective complexes is enveloping over any ring.