ON INFINITE GOLDIE-DIMENSION

Two elements x and y of a partially ordered set P are said to be disjo int if there is no z is an element of P such that z less than or equal to x and z less than or equal to y. Denote by delta(P) the supremum o f the cardinals kappa such that P contains a subset of pairwise disjoi nt elements with cardinal number kappa. P. Erdos and A. Tarski (Ann. o f Math. 44, 1943, 315-329) proved that, unless delta(P) is weakly inac cessible, P contains a subset of pairwise disjoint elements with cardi nal number delta(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 t he supremum of all cardinals kappa such that M contains the direct sum of kappa nonzero submodules. They proved that, unless Gd M is weakly inaccessible, M contains a direct sum of Gd M submodules. In this pape r, a unified proof of these two results is given. It is also shown tha t similar results hold in the context of modular lattices and abelian categories. (C) 1998 Academic Press.