## Note on simple integral extension domains and maximal chains of prime ideals

HTML articles powered by AMS MathViewer

- by L. J. Ratliff PDF
- Proc. Amer. Math. Soc.
**77**(1979), 179-185 Request permission

## Abstract:

It is shown that if*R*is a semi-local (Noetherian) domain, then there exists a simple integral extension domain $R[e]$ of

*R*such that there exists a maximal chain of prime ideals of length

*n*in some integral extension domain of

*R*if and only if there exists a maximal chain of prime ideals of length

*n*in $R[e]$. An interesting existence theorem on a certain type of height one prime ideals in $R[X]$ follows.

## References

- Evan G. Houston and Stephen McAdam,
*Chains of primes in Noetherian rings*, Indiana Univ. Math. J.**24**(1974/75), 741–753. MR**360566**, DOI 10.1512/iumj.1975.24.24057 - Stephen McAdam,
*On taut-level $R\langle x\rangle$*, Duke Math. J.**42**(1975), no. 4, 637–644. MR**387272** - Masayoshi Nagata,
*Local rings*, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR**0155856** - L. J. Ratliff Jr.,
*Four notes on saturated chains of prime ideals*, J. Algebra**39**(1976), no. 1, 75–93. MR**399072**, DOI 10.1016/0021-8693(76)90062-4 - L. J. Ratliff Jr. and S. McAdam,
*Maximal chains of prime ideals in integral extension domains. I*, Trans. Amer. Math. Soc.**224**(1976), 103–116. MR**437513**, DOI 10.1090/S0002-9947-1976-0437513-3 - L. J. Ratliff Jr. and S. McAdam,
*Maximal chains of prime ideals in integral extension domains. I*, Trans. Amer. Math. Soc.**224**(1976), 103–116. MR**437513**, DOI 10.1090/S0002-9947-1976-0437513-3 - L. J. Ratliff Jr.,
*Maximal chains of prime ideals in integral extension domains. III*, Illinois J. Math.**23**(1979), no. 3, 469–475. MR**537802** - Louis J. Ratliff Jr.,
*Chain conjectures in ring theory*, Lecture Notes in Mathematics, vol. 647, Springer, Berlin, 1978. An exposition of conjectures on catenary chains. MR**496884**
—,

*On maximal ideals and simple integral extension rings*, preprint.

## Additional Information

- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**77**(1979), 179-185 - MSC: Primary 13B20; Secondary 13A15
- DOI: https://doi.org/10.1090/S0002-9939-1979-0542081-X
- MathSciNet review: 542081