On Infinite Goldie Dimension

Two elementsxandyof a partially ordered setPare said to bedisjointif there is noz Psuch thatz ≤ xandz ≤ y. Denote by δ(P) the supremum of the cardinals κ such thatPcontains a subset of pairwise disjoint elements with cardinal number κ. P. Erdös and A. Tarski (Ann. of Math.44, 1943, 315–329) proved that, unless δ(P) is weakly inaccessible,Pcontains a subset of pairwise disjoint elements with cardinal number δ(P). J. Dauns and L. Fuchs (J. Algebra115, 1988, 297–302) defined theGoldie dimensionof a moduleM, denoted by Gd M, as the supremum of all cardinals κ such thatMcontains the direct sum of κ nonzero submodules. They proved that, unless Gd Mis weakly inaccessible,Mcontains a direct sum of Gd Msubmodules. 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.