They are too old for Math Reviews, but I think the articles in question are:

 - Von Neumann: "Zur Prüferischen Theorie der idealen Zahlen", Acta Scientiarum Mathematicum (Szeged) 2:4 (1926) (can be read online at <a href="http://acta.fyx.hu/acta/showCustomerVolume.action?id=5089&dataObjectType=volume&noDataSet=true&style=">the journal's website</a>)

 - Prüfer: "Neue Begründung der algebraischen Zahlentheorie", Math. Annalen 94 (1925), 198-243 (a link to volume 94 of the journal is at the Göttingen archive <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0094">here</a>)

in both of which one main idea seems to be (in modern language) to consider the embedding of a ring of integers $\frak{o}$ into the product $ \prod_{\frak{p},n}\frak{o}/\frak{p}^n$. The Von Neumann paper even mentions the $p$-adics. That's about all I could extract at a glance, my German being virtually nonexistent - someone with better German will be able do a more thorough job. 

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EDIT (after further reading):

The aim of both papers appears to be to develop a theory of "Dedekind ideal numbers" in which they appear as elements of an actual ring. The essential difference (in modern language) is that Prüfer uses the algebraic definition of the profinite completion of the ring of integers, whereas Von Neumann takes as his starting point the completion of the number field with respect to the product of the $p$-adic topologies. (So his ring of adeles is simply the product of the finite completions of the number fields, with the product topology). Both authors spend most of the time proving basic algebaric/topological facts about these rings. I could find no significant arithmetic applications in either paper, although Von Neumann appears to promise a sequel (never published) in which he looks at adeles of $\overline{\mathbb{Q}}$ rather than of a fixed number field, and uses them to prove a "unique factorisation" for Dedekind ideal numbers.