## The gluing of maximal ideals—spectrum of a Noetherian ring—going up and going down in polynomial rings

HTML articles powered by AMS MathViewer

- by Ada Maria de Souza Doering and Yves Lequain PDF
- Trans. Amer. Math. Soc.
**260**(1980), 583-593 Request permission

## Abstract:

If ${M_1}, ... , {M_s}$ are maximal ideals of a ring*R*that have isomorphic residue fields, then they can be “glued” in the sense that a subring

*D*of

*R*with

*R*is integral over

*D*and ${M_1} \cap D = ... = {M_s} \cap D$ can be constructed. We use this gluing process to prove the following result: Given any finite ordered set $\mathcal {B}$, there exists a reduced Noetherian ring

*B*and an embedding $\psi : \mathcal {B} \to Spec B$ such that $\psi$ establishes a bijection between the maximal (respectively minimal) elements of $\mathcal {B}$ and the maximal (respectively minimal) prime ideals of

*B*and such that given any elements $\beta ’$, $\beta ''$ of $\mathcal {B}$, there exists a saturated chain of prime ideals of length

*r*between $\psi (\beta ’)$ and $\psi (\beta '')$ if and only if there exists a saturated chain of length

*r*between $\beta ’$ and $\beta ''$. We also use the gluing process to construct a Noetherian domain

*A*with quotient field

*L*and a Noetherian domain

*B*between

*A*and

*L*such that: $A \hookrightarrow B$ possesses the Going Up and the Going Down properties, $A[X] \hookrightarrow B[X]$ is unibranched and $A[X] \hookrightarrow B[X]$ possesses neither the Going Up nor the Going Down properties.

## References

- M. F. Atiyah and I. G. Macdonald,
*Introduction to commutative algebra*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR**0242802** - Jeffrey Dawson and David E. Dobbs,
*On going down in polynomial rings*, Canadian J. Math.**26**(1974), 177–184. MR**330152**, DOI 10.4153/CJM-1974-017-9 - Ada Maria de Souza Doering,
*Chains of prime ideals in Noetherian domains*, J. Pure Appl. Algebra**18**(1980), no. 1, 97–109. MR**578571**, DOI 10.1016/0022-4049(80)90121-8 - Paul M. Eakin Jr.,
*The converse to a well known theorem on Noetherian rings*, Math. Ann.**177**(1968), 278–282. MR**225767**, DOI 10.1007/BF01350720 - Raymond C. Heitmann,
*Examples of noncatenary rings*, Trans. Amer. Math. Soc.**247**(1979), 125–136. MR**517688**, DOI 10.1090/S0002-9947-1979-0517688-0 - M. Hochster,
*Prime ideal structure in commutative rings*, Trans. Amer. Math. Soc.**142**(1969), 43–60. MR**251026**, DOI 10.1090/S0002-9947-1969-0251026-X - William J. Lewis,
*The spectrum of a ring as a partially ordered set*, J. Algebra**25**(1973), 419–434. MR**314811**, DOI 10.1016/0021-8693(73)90091-4 - Stephen McAdam,
*Going down in polynomial rings*, Canadian J. Math.**23**(1971), 704–711. MR**280482**, DOI 10.4153/CJM-1971-079-5 - Stephen McAdam,
*Going down*, Duke Math. J.**39**(1972), 633–636. MR**311658** - Stephen McAdam,
*Saturated chains in Noetherian rings*, Indiana Univ. Math. J.**23**(1973/74), 719–728. MR**332768**, DOI 10.1512/iumj.1974.23.23060 - Masayoshi Nagata,
*On the chain problem of prime ideals*, Nagoya Math. J.**10**(1956), 51–64. MR**78974**, DOI 10.1017/S0027763000000076 - 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. 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

## Additional Information

- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**260**(1980), 583-593 - MSC: Primary 13E05
- DOI: https://doi.org/10.1090/S0002-9947-1980-0574801-5
- MathSciNet review: 574801