I-adic拓扑与希尔伯特basis定理的2个推论

摘要

首先证明交换环关于I-adic拓扑诱导的完备化环是商环的有限投射极限，其次揭示Abel范畴noetherian对象与正合序列的关系，进而用范畴的方法给出希尔伯特basis定理的推论——noetherian环的I-adic拓扑完备化环R也是noetherian环的新证明，之后论证该推论与希尔伯特basis定理的另一推论是等价的．%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 investiga- ted.