Abstract: Let R be a ring. An elemente in R is called a quasi-idempotent if there is a central unit k of R such that e²=ke. If for every element a of R, there exists a quasi-idempotent e∈R and a quasinilpotent q∈R^qnil such that e∈comm²(a) and a=e+q,then R is called a strongly quasinil quasi-clean ring. If for every element a of R, there exists a quasi-idempotent e∈R such that e∈comm²(a),a+e∈U(R) and ae∈R^qnil,then R is called a quasi-quasi-polar ring. In this paper, it is proved that the quasi-quasi-polar ring and quasi-polar ring are equivalent; every strongly nil quasi-clean ring is a strongly quasinil quasi-clean ring; and every strongly quasinil quasi-clean ring is a quasi-polar ring. The inverses are not hold.

Key words: quasi-idempotent, strongly quasinil clean ring, quasi-quasi-polar ring, strongly quasinil quasi-clean ring, quasi-polar ring