Abstract: Finitely extending property is shown to carry over to matrix rings, and non-singular finitely extending rings are characterized via PP rings, semihereditary rings and finitely co-non-singular rings, these extend [2, Theorem 2.1]; a partial positive answer is provided to the FGF and CF Conjectures, that is, every right finitely extending right CF ring is right CEP (so right aritian), which improve [6, Theorem 3.7].

Key words: n-extending ring, n-co-non-singular ring, Semihereditary ring, PP ring, QF ring