DUALLY SLENDER MODULES AND STEADY RINGS

A module M over a ring R is called dually slender if Hom(R)(M,-) commu tes with direct sums of R-modules. For example, any finitely generated module is dually slender. A ring R is called right steady if each dua lly slender right R-module is finitely generated. We provide a model t heoretic necessary and sufficient condition for a countable ring to be right steady. Also, we prove that any right semiartinian ring of coun table Loewy length is right steady. For each uncountable ordinal sigma , we construct examples of commutative semiartinian rings T-sigma, and Q(sigma), of Loewy length sigma+1 such that T-sigma is, but Q(sigma) is not, steady. Finally, we study relations among dually slender, redu cing, and almost free modules.