On Infinite Goldie Dimension

✍ Scribed by Catarina Santa-Clara; Fernando C Silva

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 142 KB
• Volume: 205
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1997.7404

✦ Synopsis

Two elements x and y of a partially ordered set P are said to be disjoint if there Ž . is no z g P such that z F x and z F y. Denote by ␦ P the supremum of the cardinals such that P contains a subset of pairwise disjoint elements with Ž . cardinal number . P. Erdos and A. Tarski Ann. of Math. 44, 1943, 315᎐329 Ž . proved that, unless ␦ P is weakly inaccessible, P contains a subset of pairwise Ž . Ž disjoint elements with cardinal number ␦ P . J. Dauns and L. Fuchs J. Algebra . 115, 1988, 297᎐302 defined the Goldie dimension of a module M, denoted by Gd M, as the supremum of all cardinals such that M contains the direct sum of nonzero submodules. They proved that, unless Gd M is weakly inaccessible, M contains a direct sum of Gd M submodules. In this paper, a unified proof of these two results is given. It is also shown that similar results hold in the context of modular lattices and abelian categories.