Crawley and Jónsson introduced an exchange property for modules: An R- module M (R associative ring with identity) is said to have the (finite) exchange property if the following holds: whenever M occurs as a direct summand of a (finite) direct sum \(A=\oplus_{i\in I}A_ i\), then \(A=M\oplus \oplus_{i\in I}C_ i\) for suitale submodules \(C_ i\) of the \(A_ i.\)
One of the problems left open by Crawley and Jónsson is the following one: Does the finite exchange property imply the unrestricted exchange property? This problem was studied by Harada, Ishii, Sai, Kanbara and later also by Yamagata.
We list the relevant papers: M. Harada, Osaka J. Math. 8, 309-321 (1971; Zbl 0248.18019); M. Harada and T. Ishii, ibid. 12, 483-491 (1975; Zbl 0325.16018); M. Harada and Y. Sai, ibid. 7, 323-344 (1970; Zbl 0248.18018); H. Kanbara, ibid. 9, 409-413 (1972; Zbl 0251.16019); K. Yamagata, Pac. J. Math. 55, 301-317 (1974; Zbl 0283.16015), Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 12, 149- 158 (1974; Zbl 0307.16017); ibid. 13, 1-6 (1975; Zbl 0335.16018).
In general, finite direct sums of modules with the exchange property inherit this property. For infinite sums this is no longer true. All of the above mentioned papers have to do with comparison of three conditions:
(1) M has the exchange property.
(2) M has the finite exchange property.
(3) The family \((M_ j)_{j\in J}\) is semi-T-nilpotent, where \(M=\oplus_{j\in J}M_ j\) such that each \(M_ j\) has a local endomorphism ring.
The equivalence of (2)\(\Leftrightarrow(3)\) is proved and (3)\(\Rightarrow(1)\) is shown in the special case that all \(M_ j's\) are injective, resp. all \(M_ j's\) are isomorphic.
The authors, in this paper, show the equivalence of (1)\(\Leftrightarrow(3)\) in general. This is a consequence of their main result, which may be stated as follows: Theorem. Let \((M_ j)_{j\in J}\) be a semi-T-nilpotent family of modules (not necessarily indecomposable) with the exchange property. Then \(\oplus_{j\in J}M_ j\) has the exchange property in either of the following cases:
(I) \(M_ j\) and \(M_ k\) have no nontrivial isomorphic direct summand for \(j\neq k.\)
(II) \(M_ j\cong M_ k\) for all j,\(k\in J.\)
The proof uses a transfinite induction argument. As a corollary to this result they show: Suppose \(M=\oplus_{j\in J}M_ j\), where each \(M_ j\) is indecomposable. Then the above conditions (1), (2) and (3) are equivalent. Another major result in the present paper reads: Each strongly invariant submodule of an arbitrary algebraically compact module has the exchange property. This class of modules includes the quasi- injective modules [cf. R. B. Warfield jun., Pac. J. Math. 31, 263- 276 (1969; Zbl 0185.041)]. It also includes the \({\mathbb{Z}}\)-adically complete abelian groups [cf. R. B. Warfield jun., ibid. 34, 237-255 (1970; Zbl 0206.034)]. Finally, the torsion-complete primary abelian groups are a special case; these were considered by Crawley and Jónsson.
But there are several classes of examples covered by the statement which are new. Warfield has shown that each linearly compact module over a commutative ring is algebraically compact. So the class of all linearly compact modules over a commutative ring is another example. As a special case of the latter one can consider the class of all artinian modules over a commutative ring. In the commutative case this enables the authors to answer in the positive a question of Crawley and Jónsson whether each artinian module has the exchange property.
In giving this rather extensive review I want to indicate how the authors were able to subsume numerous occurrences of the exchange property scattered in the literature. Their nice results have a strong unifying effect.