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Journal of the London Mathematical Society 1985 s2-32(3):404-410; doi:10.1112/jlms/s2-32.3.404
© 1985 by London Mathematical Society
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© Oxford University Press

Amao's Theorem and Reduction Criteria

D. Rees

6 Hillcrest Park, Exeter EX4 4SH

In a paper written in 1974, Amao proved that if I, J are ideals of a local ring such that J {subseteq} I and l(J:I) < {infty} then, for large n, l(In/Jn) is a polynomial µ(n) in n. It is shown in the present paper that the degree of this polynomial is at most the dimension d of the local ring and, further, if the local ring is quasi-unmixed, then J is a reduction of I if and only if µ(n) has degree lessthan d. This generalises an earlier result of the author and a further generalisation is given, this time of a result of Böger which generalised the author's earlier result.


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