Steadiness is tested by a single module.

*(English)*Zbl 0988.16003
Kelarev, A. V. (ed.) et al., Abelian groups, rings and modules. Proceedings of the AGRAM 2000 conference, Perth, Australia, July 9-15, 2000. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 273, 301-308 (2001).

A module \(M\) is called dually slender if the functor \(\operatorname{Hom}(M,-)\) 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.

As 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 \(R^*=\operatorname{Hom}(R,\mathbb{Q}/\mathbb{Z})\) 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.

For the entire collection see [Zbl 0960.00043].

As 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 \(R^*=\operatorname{Hom}(R,\mathbb{Q}/\mathbb{Z})\) 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.

For the entire collection see [Zbl 0960.00043].

Reviewer: Paul Eklof (Irvine)

##### MSC:

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |

16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |

16P99 | Chain conditions, growth conditions, and other forms of finiteness for associative rings and algebras |

16E50 | von Neumann regular rings and generalizations (associative algebraic aspects) |

13C13 | Other special types of modules and ideals in commutative rings |

13E99 | Chain conditions, finiteness conditions in commutative ring theory |