Nil-quasi-clean matrix rings

Let [Formula: see text] be a ring with identity. An element [Formula: see text] in [Formula: see text] is called a quasi-idempotent if [Formula: see text] for some central unit [Formula: see text] in [Formula: see text]. An element [Formula: see text] in [Formula: see text] is called a nil-quasi-clean element if [Formula: see text] is a sum of a quasi-idempotent and a nilpotent element. If every element of [Formula: see text] is nil-quasi-clean, then [Formula: see text] is called a nil-quasi-clean ring. Let [Formula: see text] be a field. This paper studies the nil-quasi-cleanness of the matrix ring [Formula: see text]. It is proved that, for [Formula: see text], [Formula: see text] is nil-quasi-clean if and only if [Formula: see text] or [Formula: see text] is a complete field of characteristic two and [Formula: see text]. This gives a partial answer of an open question in the reference [G. Tang and Y. Zhou, Nil [Formula: see text]-cleanness and strongly nil [Formula: see text]-cleanness of rings, J. Algebra Appl. 21(4) (2022) 2250077].