THE LOEWY SERIES OF AN FCP (DISTRIBUTIVE) RING EXTENSION

Authors : Gabriel PICAVET, Martine PICAVETL`HERMITTE

Pages : 15-49

Doi:10.24330/ieja.851985

View : 15 | Download : 17

Publication Date : 2021-01-05

Article Type : Research Paper

Abstract :If $R\subseteq S$ is an extension of commutative rings, we consider the lattice $insert ignore into journalissuearticles values([R,S],\subseteq);$ 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