×

The set of intermediate rings as a Boolean algebra. (English) Zbl 1317.13015

Let \(R\subseteq S\) be a ring extension of commutative integral domains with 1. Denote by \([R,S]\) the set of intermediate rings between \(R\) and \(S\). We say that \(R\subseteq S\) is a minimal extension if \([R,S]=\{R,S\}.\) We say that \(R=R_0\subseteq R_1\subseteq\ldots\subseteq R_n=S\) is a finite maximal chain of length \(n\) if \([R_i,R_{i+1}]\) is a minimal extension for each \(i=0,\ldots ,n-1\). The main result of the paper is the following: Assume that the extension \(R\subseteq S\) has a finite maximal chain of length \(n\) ad that the support of \(S\) as an \(R-\)module consists of \(n\) maximal ideals. Then \(([R,S], \cdot, \cap)\) has a structure of a boolean algebra of cardinality \(2^n\), where \(H\cdot G\) means the smallest subring of \(S\) containing \(H\cup G.\) Some interesting consequences of this result are proved and some nice examples are discussed.

MSC:

13B02 Extension theory of commutative rings
06E25 Boolean algebras with additional operations (diagonalizable algebras, etc.)
13G05 Integral domains
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Atiyah M. F., Introduction to Commutative Algebra (1969) · Zbl 0175.03601
[2] DOI: 10.1081/AGB-120024849 · Zbl 1029.13004
[3] Ayache A., Inter. Elec. J. Algebra 8 pp 129– (2010)
[4] DOI: 10.1080/00927872.2012.706842 · Zbl 1288.13006
[5] Bourbaki N., Algèbre Commutative (1961)
[6] Coykendall J., J. Algebra, Number Theory. Appl. 13 (2) pp 121– (2009)
[7] DOI: 10.1080/00927870802254546 · Zbl 1159.13011
[8] Dobbs D. E., Inter. Elec. J. Alg. 5 pp 121– (2009)
[9] DOI: 10.1016/j.jalgebra.2012.07.055 · Zbl 1271.13022
[10] Dobbs D. E., J. Algebra. Number Theory Appl. 26 pp 103– (2012)
[11] DOI: 10.1016/0021-8693(70)90020-7 · Zbl 0218.13011
[12] Gilmer R., Multiplicative Ideal Theory (1972) · Zbl 0248.13001
[13] DOI: 10.1080/00927879908826495 · Zbl 0972.13008
[14] DOI: 10.1007/978-0-387-36717-0_22
[15] DOI: 10.1080/00927879208824427 · Zbl 0747.13017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.