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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].

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