A new class of nil-clean elements which are exchange

摘要

An element a in a ring R is called left medium nil-clean if a = e +t with idempotent e and nilpotent t, such that et and t commute. This proper class of nil-clean elements properly includes the strongly nil-clean elements.We show that in any unital ring, left medium nil-clean elements are exchange. Over projective-free domains we show that 2 × 2 and 3 × 3 left (or right) medium nil-clean matrices are strongly nil-clean.