For an arbitrary infinite cardinal κ, we define classes of κ-cslender and κ-tslender modules as well as related classes of κ-hmodules and initiate a study of these classes.
We prove a unique decomposition theorem for direct products of finitely generated modules over certain classes of rings, which is analogous to the classical Krull–Schmidt–Remak–Azumaya theorem for ...direct-sum decompositions of modules.
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Criteria are obtained for a filter F of subsets of a set I to be an intersection of finitely many ultrafilters. respectively, finitely many κ-complete ultrafilters for a given uncountable cardinal ...which yield quick proofs of some results in the literature: the Los-Eda theorem (characterizing homomorphisms from a not-necessarily-countable direct product of modules to a slender module), and some results of Nablus and the author on homomorphisms on infinite direct products of not-necessarily-associative κ-algebras. The same tools allow other results of Nahlus and the author to be nontrivially strengthened, and yield an analog to one of their results, with nonabelian groups taking the place of κ-algebras. We briefly examine the question of how the common technique used in applying the general results of this note to κ-algebras on the one hand, and to nonabelian groups on the other, might be extended to more general varieties of algebras in the sense of universal algebra. In a final section, the Erdös-Kaplansky theorem on dimensions of vector spaces D¹ (D a division ring) is extended to reduced products D¹/F. and an application is noted.
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