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Viene studiato il completamento di Jacobson di uno spazio topologico , e ne vengono fatte applicazioni allo studio degli anelli comutativi con identità.
Si studiano gli anelli di funzioni continue definite su uno spazio topologico di Hausdorff O-dimensionale a valori in un campo completo F non archimedeo di rango uno, a valutazione non banale.
Si studia l'algebra topologica F(T) delle funzioni continue che applicano uno spazio O-dimensionale T in un campo valutato non archimedeo completo, munito della topologia compatto-aperta.
Vedi la Nota III, dallo stesso titolo, di cui la presente è il completamento.
In questa nota X denota un'algebra topologica Hausdorff commutativa, completa, complessa, localmente m-convessa con unità. Si ottengono condizioni sotto le quali le derivazioni di X proiettano X dentro il radicale di X. Inoltre si danno criteri secondo i quali X è un'algebra di Banach. È dimostrato che per Q-algebre imbarilate ("barreled") X, se è una collezione aperta nel piano complesso, allora l'estensione principale di in X è aperta. È anche dimostrato che un semplicemente connesso è permanente...
Si studia l'algebra topologica delle funzioni continue che applicano uno spazio -dimensionale in un campo valutato non archimedeo completo, munito della topologia compatto-aperta.
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