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

JOURNAL OF ALGEBRA

卷:308

Article

Rush, David E.

关键词: projectively equivalent;    Rees valuations;    multiplicative lattice;    Krull lattice;    Mori-Nagata theorem;    A-transform;    pseudo-valuation;   

DOI  :  10.1016/j.jalgebra.2006.08.014

来源: Elsevier

【 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, A is an element of L, {P(l),..., P(r)} is the set of centers of the Rees valuations v(l),..., v(r) of A and e is the least common multiple of the Rees numbers e(l) (A),..., e(r)(A) of A, then Ass(L/A(n/e)) subset of {P(l),..., P(r)}, where A = V{x is an element of L vertical bar (v) over bar (A)(x) >= beta}. Further, if A not less than or equal to q for each minimal prime q is an element of L, then Ass(L/A(n/e)) subset of Ass(L/A(n/e+k/e)) for each n is an element of N, where k/e is in a certain additive subsernigroup of Q(+) 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 LT Ratliff who gave them for rings in the case that the n/e and k/e are integers. (c) 2006 Elsevier Inc. All rights reserved.

【 授权许可】

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