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On some intermediate rings
Published online by Cambridge University Press: 26 February 2010
Extract
Let R be a commutative, Noetherian ring and let Q be the total quotient ring of R. We shall call B an intermediate ring if R ⊂ B ⊂ Q. In [S] it is proved, for an integral domain R, that if R ⊂ B ⊂ Rf where B is flat over R, then B is a finitely generated R-algebra. We observe that the result holds for any commutative, Noetherian ring where f is a non-zero divisor. Our proof [Theorem 1.1] is a little different and straight; it is given for completeness. The idea of the proof in [S] lies in finding an ideal I of R such that IB = B, and for any λ∈I, b∈B there exists m ≥ 1 such that λmb ∈ R. We shall show that even if an intermediate ring B is finitely generated R-algebra, there may not exist any ideal I of R such that IB = B, moreover, if B is not finitely generated R-algebra, we may have IB = B for some ideal I in R.
MSC classification
Secondary:
13E05: Noetherian rings and modules
- Type
- Research Article
- Information
- Copyright
- Copyright © University College London 1992
References
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