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Products of small modules. (English) Zbl 1307.16006
The paper gives a partial answer (depending on the model of set theory) to the problem of existence of rings such that the class of all small modules \(M\) (i.e., the covariant functor \(\operatorname{Hom}(M,-)\) commutes with all direct sums) is closed under direct products. In Proposition 2.3, it is shown that these considerations can be restricted to the case of simple self-injective regular rings. The main result of the paper (Theorem 3.5) says that if \(R\) is a non-Artinian right self-injective right purely infinite (i.e., there is a right ideal \(K\) of \(R\) such that \(K\simeq R^{(\omega)}\) as right \(R\)-modules) associative ring with unit then the class of all small right \(R\)-modules is closed under direct products under the assumption (which is consistent with ZFC) that there is no strongly inaccessible cardinal (i.e., a regular cardinal \(\kappa\) such that \(2^\lambda<\kappa\) for each \(\lambda<\kappa\)).

16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16B70 Applications of logic in associative algebras
03E35 Consistency and independence results
16D50 Injective modules, self-injective associative rings
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
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