Abstract: This paper first shows that the completion of a commutative ring induced by I-adic topology is the finite projective limit of its quotient rings.And then the relationship between noetherian objects and exact sequences is given;moreover,a new proof for the lemma of Hilbert basis theorem is presented，which states that：the completion of a noetherian ring induced by I-adic topology is again a noetherian ring.Also，the equivalence of the above lemma and the other lemma of Hilbert basis theorem is investigated.

Key words: I-adic topology, completion, formal power series ring, noetherian ring