Steadiness is tested by a single module (Q2738766)

A module \(M\) is called dually slender if the functor \(\Hom(M,-)\) commutes with direct sums. A ring \(R\) is called (right) steady if the only dually slender (right) \(R\)-modules are the finitely-generated ones. Various finiteness conditions on a ring, for example being Noetherian, imply that a ring is steady, but a ring-theoretic characterization is not known.NEWLINENEWLINENEWLINEAs indicated by the title of this paper, the emphasis here is on characterizing the steadiness of a ring in terms of the non-existence of counterexamples inside a particular module. The main theorem states that a commutative regular ring \(R\) is steady if and only if the right \(R\)-module \(R^*=\Hom(R,\mathbb{Q}/\mathbb{Z})\) does not contain an infinitely generated dually slender module. For any commutative ring \(R\), it is proved that if \(R^*\) does not contain an infinitely generated dually slender module, then there is only a set, i.e., not a proper class, of pairwise non-isomorphic counterexamples to the ring being steady.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00043].