Ultraproducts of PM-rings and MP-rings
The author considers properties of commutative rings preserved under ultraproducts. Since an ultraproduct is a quotient of a direct product it is sufficient to treat these two constructions separately. The properties considered are PM (every prime is contained in a unique maximal), MP (every prime contains a unique minimal), soft (Hausdorff maximal spectrum and zero Jacobson radical), TB (maximal spectrum homeomorphic to Boolean spectrum), weak Baer (element annihilators are idempotent generated principal), and Baer (ideal annihilators are idempotent generated principal). All are shown to be preserved by ultra-products for Baer and MP, and the latter is preserved for reduced rings. One technical tool, used in the Baer result, characterizes the incomplete ultraproducts of Boolean algebras.
scientific article; zbMATH DE number 3857250
scientific article