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Localization and Completion at Primes Generated by Normalizing Sequences in Right Noetherian Rings

  • A. G. Heinicke (a1)

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If P is a right localizable prime ideal in a right Noetherian ring R, it is known that the ring RP is right Noetherian, that its Jacobson radical is the only maximal ideal, and that RP /J(RP) is simple Artinian: in short it has several properties of the commutative local rings.

In the present work we examine the properties of RP under the additional assumption that P is generated by, or is a minimal prime above, a normalizing sequence. It is shown that in such cases J(RP) satisfies the AR-property (i.e., P is classical) and that the rank of P coincides with the Krull dimension of RP. The length of the normalizing sequence is shown to be an upper bound for the rank of P, and if P is generated by a normalizing sequence x 1, x 2, …, xn then the rank of P equals n if and only if the P-closures of the ideals Ij generated by x 1, x 2, …, xj (j = 0, 1, …, n), are all distinct primes.

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References

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Localization and Completion at Primes Generated by Normalizing Sequences in Right Noetherian Rings

  • A. G. Heinicke (a1)

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