Completely centrally essential rings

A ring R is said to be centrally essential if for every its nonzero element a, there exist nonzero central elements x and y with [Formula: see text]. A ring R is said to be completely centrally essential if all its factor rings are centrally essential rings. It is proved that completely centrally essential semiprimary rings are Lie nilpotent; noetherian completely centrally essential rings are strongly Lie nilpotent (in particular, every such ring is a [Formula: see text]-ring). Every completely centrally essential ring has the classical ring of fractions which is a completely centrally essential ring. If R is a commutative domain and G is an arbitrary group, then any completely centrally essential group ring [Formula: see text] is commutative.