Products of small modules.

*(English)*Zbl 1307.16006The 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 \(\lambda<\kappa\)).

Reviewer: Petr Němec (Praha)

##### MSC:

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) |