It is well known that projective
modules P are characterized by the property that each surjection M → P of modules
splits. For arbitrary modules A one can ask for conditions under which each
surjection A(c)→ A(d) will split where c and d are cardinals. Modules with this
property are said to have the Baer splitting property. If the surjection A(c)→ A(d)
splits whenever d is a finite cardinal then A is said to have the finite Baer splittingproperty. If the surjection A(c)→ A(d) splits whenever c and d are finite cardinals
then A is said to have the endlich Baer splitting property. Albrecht generalizes a
theorem of Arnold and Lady by showing that if A satisfies mild hypotheses,
then A has the Baer splitting property iff IA≠A for each proper right ideal
I ⊂End(A).
The goal of this paper is to organize what is known about the (finite, endlich)
Baer splitting property by generalizing to pairs (A,P) that have the (endlich) Baer
splitting property. (See definitions below.) As an application, we show that the
torsion-free abelian group of finite rank has the finite Baer splitting property iff it has
the endlich Baer splitting property. We cite examples to show that this result is not
true of countable modules.
