## Catenarity in module-finite algebras

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- by Shiro Goto and Kenji Nishida PDF
- Proc. Amer. Math. Soc.
**127**(1999), 3495-3502 Request permission

## Abstract:

The main theorem says that any module-finite (but not necessarily commutative) algebra $\Lambda$ over a commutative Noetherian universally catenary ring $R$ is catenary. Hence the ring $\Lambda$ is catenary if $R$ is Cohen-Macaulay. When $R$ is local and $\Lambda$ is a Cohen-Macaulay $R$-module, we have that $\Lambda$ is a catenary ring, $\dim \Lambda =\dim \Lambda /Q+\mathrm {ht}_\Lambda Q$ for any $Q\in \operatorname {Spec}\Lambda$, and the equality $n=\mathrm {ht}_\Lambda Q- \mathrm {ht}_\Lambda P$ holds true for any pair $P\subseteq Q$ of prime ideals in $\Lambda$ and for any saturated chain $P=P_0\subset P_1\subset \cdots \subset P_n=Q$ of prime ideals between $P$ and $Q$.## References

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## Additional Information

**Shiro Goto**- Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, Kawasaki 214-71, Japan
- MR Author ID: 192104
- Email: goto@math.meiji.ac.jp
**Kenji Nishida**- Affiliation: Department of Mathematics, Faculty of Science, Shinsyu University, Matsumoto, 390-0802 Japan
- Email: kenisida@math.shinsyu-u.ac.jp
- Received by editor(s): October 27, 1997
- Received by editor(s) in revised form: February 24, 1998
- Published electronically: May 13, 1999
- Additional Notes: The first author was supported by the Grant-in-Aid for Scientific Researches (C)
- Communicated by: Ken Goodearl
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**127**(1999), 3495-3502 - MSC (1991): Primary 13E05, 16A18; Secondary 13H10, 16A33
- DOI: https://doi.org/10.1090/S0002-9939-99-04962-X
- MathSciNet review: 1618674