THE LOEWY SERIES OF AN FCP (DISTRIBUTIVE) RING EXTENSION

Year 2021, Volume: 29 Issue: 29, 15 - 49, 05.01.2021

Gabriel Pıcavet Martine Pıcavet-l'hermıtte

https://doi.org/10.24330/ieja.851985

Cited By: 2

Abstract

If $R\subseteq S$ is an extension of commutative rings, we consider the lattice $([R,S],\subseteq)$ of all the $R$-subalgebras of $S$.
We assume that the poset $[R,S]$ is both Artinian and Noetherian; that is, $R\subseteq S$ is an FCP extension.
The Loewy series of such lattices are studied. Most of main results are gotten in case these posets are distributive,
which occurs for integrally closed extensions. In general, the situation is much more complicated. We give a discussion for finite field extensions.

Keywords

FIP, FCP extension, minimal extension, support of a module, distributive lattice, Boolean lattice, atom, socle, Loewy series, Galois extension

References

• M. Ben Nasr and N. Jarboui, New results about normal pairs of rings with zero divisors, Ric. Mat., 63(1) (2014), 149-155.
• N. Bourbaki; Algebre Commutative, Chs. 1-2, Hermann, Paris, 1961.
• N. Bourbaki, Algebre, Chs. 4-7, Masson, Paris, 1981.
• P.-J. Cahen, G. Picavet and M. Picavet-L'Hermitte, Pointwise minimal extensions, Arab. J. Math. (Springer), 7(4) (2018), 249-271.
• G. Calugareanu, Lattice Concepts of Module Theory, Kluwer Academic Publishers, Dordrecht, 2000.
• D. E. Dobbs, B. Mullins, G. Picavet and M. Picavet-L'Hermitte, On the FIP property for extensions of commutative rings, Comm. Algebra, 33(9) (2005), 3091-3119.
• 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.
• D. E. Dobbs, G. Picavet, M. Picavet-L'Hermitte and J. Shapiro, On intersections and composites of minimal ring extensions, JP J. Algebra Number Theory Appl., 26 (2012), 103-158.
• D. E. Dobbs, G. Picavet and M. Picavet-L'Hermitte, When an extension of Nagata rings has only finitely many intermediate rings, each of those is a Nagata ring, Int. J. Math. Math. Sci, (2014), 315919 (13 pp).
• D. E. Dobbs, G. Picavet and M. Picavet-L'Hermitte, Transfer results for the FIP and FCP properties of ring extensions, Comm. Algebra, 43 (2015), 1279- 1316.
• D. Ferrand and J.-P. Olivier, Homomorphismes minimaux d'anneaux, J. Algebra, 16 (1970), 461-471.
• R. Gilmer and W. Heinzer, On the existence of exceptional field extensions, Bull. Amer. Math. Soc., 74 (1968), 545-547.
• G. Gratzer, General Lattice Theory, Academic Press, New York-London, 1978.
• M. Hall, The Theory of Groups, The Macmillan Co., New York, 1959.
• J. A. Huckaba and I. J. Papick, A note on a class of extensions, Rend. Circ. Mat. Palermo (2), 38 (1989), 430-436.
• N. Jarboui, A note on the (FMC) condition for extensions of commutative rings, Int. J. Open Problems Comput. Math., 5(3) (2012), 88-95.
• M. Knebusch and D. Zhang, Manis Valuations and Prufer Extensions I, Springer-Verlag, Berlin, 2002.
• S. Mac Lane and G. Birkhoff, Algebra, Amer. Math. Soc., 1999.
• P. Morandi, Field and Galois Theory, Springer-Verlag, New York, 1996.
• C. Nastasescu and F. Van Oystaeyen, Dimensions of Ring Theory, Mathematics and its applications, D. Reidel Publishing Co., Dordrecht, 1987.
• D. G. Northcott, Lessons on Rings, Modules and Multiplicities, Cambridge University Press, London, 1968.
• G. Picavet and M. Picavet-L'Hermitte, T-Closedness, in: Non-Noetherian Commutative Ring Theory, Math. Appl. 520, Kluwer, Dordrecht, (2000), 369- 386.
• G. Picavet and M. Picavet-L'Hermitte, About minimal morphisms, in: Multiplicative Ideal Theory in Commutative Algebra, Springer, New York, (2006), 369-386.
• G. Picavet and M. Picavet-L'Hermitte, Prufer and Morita hulls of FCP extensions, Comm. Algebra, 43 (2015), 102-119.
• G. Picavet and M. Picavet-L'Hermitte, Some more combinatorics results on Nagata extensions, Palest. J. Math., 5 (2016), 49-62.
• G. Picavet and M. Picavet-L'Hermitte, Modules with finitely submodules, Int. Electron. J. Algebra, 19 (2016), 119-131.
• G. Picavet and M. Picavet-L'Hermitte, Quasi-Prufer extensions of rings, in: Rings, Polynomials and Modules, Springer, (2017), 307-336.
• G. Picavet and M. Picavet-L'Hermitte, Rings extensions of length two, J. Algebra Appl., 18(8) (2019), 1950174 (34 pp).
• G. Picavet and M. Picavet-L'Hermitte, Boolean FIP ring extensions, Comm. Algebra, 48 (2020), 1821-1852.
• G. Picavet and M. Picavet-L'Hermitte, Catenarian FCP ring extensions, to appear in J. Commut. Algebra.
• S. Roman, Lattices and Ordered Sets, Springer, New York, 2008.
• R. P. Stanley, Enumerative Combinatorics, Vol. 1, Second edition, Cambridge University Press, Cambridge, 2012.
• R. G. Swan, On seminormality, J. Algebra, 67 (1980), 210-229.