Jaffard ring
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In mathematics, a Jaffard ring is a type of ring, more general than a Noetherian ring. Formally, a Jaffard ring is a ring R such that
![\dim R[T] = 1 + \dim R, \,](../../../../math/3/c/4/3c495b722a8f4a91075938135fd3c83f.png)
where "dim" denotes Krull dimension. A Jaffard ring that is also an integral domain is called a Jaffard domain.
The Jaffard property is satisfied by any Noetherian ring R, so examples of non-Jaffardian rings are quite difficult to find. Nonetheless, an example was given in 1953 by Abraham Seidenberg: the subring of
![\overline{\mathbf{Q}} [[T]]](../../../../math/d/7/b/d7ba8e339e1cc55f1287ac434dad7a27.png)
consisting of those formal power series whose constant term is rational.
[edit] References
- Seidenberg, Abraham (1953). "A note on the dimension theory of rings". Pacific J. Math. 3: 505–512. ISSN 0030-8730. MR0054571
