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On a class of commutative Noetherian rings

  • Lindsay Burch (a1)

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If A, B are ideals of a commutative ring R, such that BA, and, for some positive integer r, Ar = BAr−1 then B is said to be a reduction of A. (This concept was defined and developed by Northcott and Rees in (1).) In this paper, I shall consider commutative Noetherian rings with the property that no non-zero principal ideal is a reduction of an ideal properly containing it.

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(1)Northcott, D. G. and Rees, D.Reductions of ideals in local rings. Proc. Cambridge Philos. Soc. 50 (1954), 145158.
(2)Zariski, O. and Samuel, P.Commutative Algebra, vol. I (D. Van Nostrand, 1958).
(3)Burch, L.On ideals of finite homological dimension in local rings. Proc. Cambridge Philos. Soc. 64 (1968), 941948.

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