| Popis: |
Two elements x and y of a partially ordered set P are said to be disjoint if there is no z ∈ P such that z ≤ x and z ≤ y . Denote by δ( P ) the supremum of the cardinals κ such that P contains a subset of pairwise disjoint elements with cardinal number κ. P. Erdos and A. Tarski ( Ann. of Math. 44 , 1943, 315–329) proved that, unless δ( P ) is weakly inaccessible, P contains a subset of pairwise disjoint elements with cardinal number δ( 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 κ such that M contains the direct sum of κ 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. |
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