Toma Albu: Completely irreducible meet decompositions in lattices, with applications to Grothendieck categories and torsion theories (I), p.393-419

Abstract:

The aim of this paper, consisting of two parts, is to investigate decompositions of elements in upper continuous modular lattices as intersections of (completely) irreducible elements. Thus, we extend from modules to such lattices the results of J. Fort [Math. Z. 103 (1967), 363-388] and consider a similar setting where irreducible submodules are replaced by completely irreducible submodules. We also extend from ideals to lattices some results of L. Fuchs, W. Heinzer, and B. Olberding [Trans. Amer. Math. Soc. 358 (2006), 3113-3131] concerning decompositions of ideals in arbitrary commutative rings as (irredundant) intersections of completely irreducible ideals. Applications of these results are given to Grothendieck categories and module categories equipped with a hereditary torsion theory.

Key Words: subdirectly irreducible poset, irreducible element, completely irreducible element, coirreducible (uniform) element, completely coirreducible element, upper continuous lattice, modular lattice, irreducible decomposition, irredundant irreducible decomposition, completely irreducible decomposition, irredundant completely irreducible decomposition, atomic lattice, weakly atomic lattice, strongly atomic lattice, semi-Artinian lattice, poset rich in subdirectly irreducibles, lattice rich in completely irreducibles, lattice rich in coirreducibles, lattice rich in completely coirreducibles, Krull dimension, dual Krull dimension, Gabriel dimension, Grothendieck category, torsion theory.

2000 Mathematics Subject Classification: Primary: 06B23, 06B35,
Secondary: 06C05, 16S90, 18E15.

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