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Nasr, Mabrouk Ben; Zeidi, Nabil

A note on the FIP property for extensions of commutative rings. (English) Zbl 1499.13030

Filomat 33, No. 19, 6213-6218 (2019).

MSC:

13B02 Extension theory of commutative rings
13A15 Ideals and multiplicative ideal theory in commutative rings
13B21 Integral dependence in commutative rings; going up, going down
13B25 Polynomials over commutative rings
13E05 Commutative Noetherian rings and modules
13E10 Commutative Artinian rings and modules, finite-dimensional algebras

Keywords:

FIP property; ring extension; intermediate ring; minimal ring extension; integral; nilpotent element
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References:

[1] D.D. Anderson, D.E. Dobbs, B. Mullins, The primitive element theorem for commutative algebra, Houston J. Math. 25 (1999) 603-623; Corrigendum: Houston J. Math. 28 (2002) 217-219. · Zbl 0999.13003
[2] D.E. Dobbs, B. Mullins, G. Picavet and M. Picavet-L’Hermitte, On the FIP property for extensions of commutative rings, Comm. Algebra 33 (2005) 3091-3119. · Zbl 1120.13009
[3] D.E. Dobbs, B. Mullins, G. Picavet andM. Picavet-L’Hermitte,On the maximal cardinality of chains of intermediate rings,Int. Electron. J.Algebra 5 (2009) 121134. · Zbl 1161.13302
[4] D.E.Dobbs, G.Picavet and M.Picavet-L’Hermitte, Characterizing the ring extensions that satisfy FIP or FCP, J.Algebra 371 (2012) 391-429. · Zbl 1271.13022
[5] D.E. Dobbs, G. Picavet and M. Picavet-L’Hermitte, Transfer results for the FIP and FCP properties of ring extensions, J. Algebra 43 (2015) 1279-1316. · Zbl 1317.13016
[6] D. Ferrand, J. P. Olivier, Homomorphismes minimaux d’anneaux, J. Algebra 16 (1970) 461-471. · Zbl 0218.13011
[7] M. S. Gilbert, Extensions of commutative rings with linearly ordered intermediate rings, Ph. D. dissertation, University of Tennessee, Knoxville, 1996.
[8] R. Gilmer, Some finiteness conditions on the set of overrings of an integral domain, Proc. Amer. Math. Soc 131 (2002) 2337-2346. · Zbl 1017.13009
[9] G. Picavet and M. Picavet-L’Hermitte, About minimal morphisms, Multiplicative Ideal Theory in Commutative Algebra, Springer-Verlag, New York, 2006 pp.369-386. · Zbl 1127.13006
[10] G. Picavet and M. Picavet-L’Hermitte, FIP and FCP products of ring morphisms, Pales. J. Math., 5 (Spec1) (2016), 63-80. · Zbl 1346.13013
[11] A. Jaballah, Ring extensions with some finiteness conditions on the set of intermediate rings, Czechoslovak Math. J. 60 (135) (2010) 117-124. · Zbl 1224.13011
[12] I. Kaplansky, Commutative Rings, rev. edn. University of Chicago Press, Chicago, 1974. · Zbl 0296.13001
[13] O. Zariski and P. Samuel, Commutative Algebra, vol. I, Van Nostrand, Princeton, 1958 · Zbl 0081.26501
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