Rees valuations and asymptotic primes of rational powers in Noetherian rings and lattices

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• 作者：David E. Rush
• 关键词：Projectively equivalent ; Rees valuations ; Multiplicative lattice ; Krull lattice ; Mori– ; Nagata theorem ; A-transform ; Pseudo-valuation
• 刊名：Journal of Algebra
• 出版年：2007
• 期刊代码：69_00218693
• 类别：mt
• 出版时间：1 February 2007
• 卷：308
• 期：1
• 页码：295-320
• 文件大小：310 K

摘要

We extend a theorem of D. Rees on the existence of Rees valuations of an ideal A of a Noetherian ring to Noetherian multiplicative lattices L. This result also extends a result of D.P. Brithinee. We then apply this to projective equivalence and asymptotic primes of rational powers of A. In particular, it is shown that if L is a Noetherian multiplicative lattice, AL, {P₁,…,P_r} is the set of centers of the Rees valuations v₁,…,v_r of A and e is the least common multiple of the Rees numbers e₁(A),…,e_r(A) of A, then Ass(L/A_n/e){P₁,…,P_r}, where . Further, if A/q for each minimal prime qL, then Ass(L/A_n/e)Ass(L/A_n/e+k/e) for each , where k/e is in a certain additive subsemigroup of which is naturally associated to the set of members of L which are projectively equivalent to A. These latter results are new even in the case of rings and extend results of L.J. Ratliff who gave them for rings in the case that the n/e and k/e are integers.