Finite exchange vs. infinite exchange

Determine whether the finite exchange property implies the infinite exchange property for rings; equivalently, decide whether every ring that satisfies exchange for finite direct-sum decompositions necessarily satisfies exchange for arbitrary (possibly infinite) decompositions in the sense of Warfield.

Background

The paper studies summation systems and their interaction with topology and quotient constructions, focusing in part on properties of reorderable rings and the behavior of Σ-closed ideals. In analyzing when Jacobson radicals remain closed under Σ-sums in quotients, the author notes a scenario where a Σ-closed ideal need not be closed in the finite topology, potentially producing quotient rings that are not Hausdorff topological rings.

In this context, the author observes that if such a scenario occurs for exchange rings (in Warfield’s sense), it would provide a negative answer to a longstanding problem of Crawley and Jónsson concerning whether the finite exchange property implies the infinite exchange property. Thus, the general implication from finite exchange to infinite exchange remains an explicit unresolved question.