Steady rings may contain large sets of orthogonal idempotents.

*(English)*Zbl 0845.16009
Facchini, Alberto (ed.) et al., Abelian groups and modules. Proceedings of the Padova conference, Padova, Italy, June 23-July 1, 1994. Dordrecht: Kluwer Academic Publishers. Math. Appl., Dordr. 343, 467-473 (1995).

Let \(R\) be a ring. A right \(R\)-module \(M\) is called small when the functor \(\text{Hom}_R(M,-)\) commutes with direct sums in the category Mod-\(R\) of right \(R\)-modules. It is well known that every finitely generated right \(R\)-module is small but the converse is not true in general; the rings for which both classes of modules coincide are called right steady.

It was known by results of R. Colpi and J. Trlifaj [Commun. Algebra 22, No. 10, 3985-3995 (1994; Zbl 0818.16003)] that a simple ring containing an infinite set of orthogonal idempotents is neither right nor left steady but the problem of whether this is true for not necessarily simple rings remained open. The main result of the paper under review provides a negative answer to this question by showing that, for each cardinal \(\kappa\), there exists a left and right steady ring containing a set of orthogonal central idempotents of cardinality \(\kappa\). In the last part of the paper, further constructions of non-steady rings are presented and it is shown that a direct product of rings is (right) steady if and only if so is each factor and the number of factors is finite.

For the entire collection see [Zbl 0830.00031].

It was known by results of R. Colpi and J. Trlifaj [Commun. Algebra 22, No. 10, 3985-3995 (1994; Zbl 0818.16003)] that a simple ring containing an infinite set of orthogonal idempotents is neither right nor left steady but the problem of whether this is true for not necessarily simple rings remained open. The main result of the paper under review provides a negative answer to this question by showing that, for each cardinal \(\kappa\), there exists a left and right steady ring containing a set of orthogonal central idempotents of cardinality \(\kappa\). In the last part of the paper, further constructions of non-steady rings are presented and it is shown that a direct product of rings is (right) steady if and only if so is each factor and the number of factors is finite.

For the entire collection see [Zbl 0830.00031].

Reviewer: J.L.Gómez-Pardo (Santiago de Compostela)

##### MSC:

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

16P20 | Artinian rings and modules (associative rings and algebras) |

16S15 | Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) |