Criteria of steadiness.

*(English)*Zbl 0917.16004
Dikranjan, Dikran (ed.) et al., Abelian groups, module theory, and topology. Proceedings in honour of Adalberto Orsatti’s 60th birthday, Padua, Italy, 1997. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 201, 359-371 (1998).

A (right) module \(M\) over a ring \(R\) is called dually slender if it is not the union of a countably infinite chain of proper submodules. H. Bass [Algebraic \(K\)-theory, Benjamin, New York (1968; Zbl 0174.30302)] noted that \(M\) is dually slender iff \(\operatorname{Hom}_R(M,-)\) commutes with direct sums. Since a module is finitely generated iff it is not the union of an infinite chain of proper submodules, finitely generated modules are dually slender. The converse is false in general and a ring \(R\) is called (right) steady if its dually slender right \(R\)-modules are finitely generated. The class of steady rings includes the noetherian rings [R. Rentschler, C. R. Acad. Sci., Paris, Sér. A 268, 930-933 (1969; Zbl 0179.06102) or R. Colpi and C. Menini, J. Algebra 158, No. 2, 400-419 (1993; Zbl 0795.16005)], perfect rings [R. Colpi and J. Trlifaj, Commun. Algebra 22, No. 10, 3985-3995 (1994; Zbl 0818.16003)], and semiartinian rings of countably Loewy length [P. C. Eklof, K. R. Goodearl and J. Trlifaj, Forum Math. 9, No. 1, 61-74 (1997; Zbl 0866.16003)].

The class of dually slender modules is occasionally much larger than that of the finitely generated ones and so it is useful to consider the following intermediate classes of modules. For any infinite cardinal \(\kappa\), a module \(M\) is called \(\kappa\)-reducing if each \(\kappa\)-generated submodule of \(M\) is contained in a finitely generated one. Then for all cardinals \(\omega<\kappa<\lambda\) we have finitely generated \(\Rightarrow\lambda\)-reducing \(\Rightarrow\kappa\)-reducing \(\Rightarrow\omega\)-reducing \(\Rightarrow\) dually slender. Eklof, Goodearl and Trlifaj (loc. cit.) have shown that, in general, these implications are irreversible. (Note that \(M\) is finitely generated iff it is \(\kappa\)-reducing for all infinite \(\kappa\).)

The aim of the paper under review is to provide a ring-theoretic criterion of steadiness for particular classes of rings and a general characterization of when all \(\omega\)-reducing modules are finitely generated. We now record the two main results, Criteria A and B.

Criterion A says that if \(R\) is a commutative semiartinian ring then \(R\) is not steady iff there is a two-sided ideal \(J\) of \(R\) containing the Jacobson radical of \(R\) and a member \(I\) of the Loewy (socle) series of the ring \(R/J\) such that \(I\) is an infinitely generated dually slender right \(R/J\)-module. Criterion B says that, letting \(M_n(R)\) denote the ring of \(n\times n\) matrices over the ring \(R\), there is an infinitely generated \(\omega\)-reducing \(R\)-module iff for some finite \(n\) there is a family \(\{A_{\alpha\beta}:\alpha<\beta<\omega_1\}\) of elements of \(M_n(R)\) and a family \(\{I_\alpha:\alpha<\omega_1\}\) of right ideals of \(M_n(R)\) such that, for all \(\alpha<\beta<\gamma<\omega_1\), (a) \(A_{\beta\gamma} A_{\alpha\beta}+I_\gamma=A_{\alpha\gamma}+I_\gamma\), (b) for all \(A\in M_n(R)\), \(A_{\alpha\gamma}A+ I_\gamma\neq A_{\beta\gamma}+ I_\gamma\) and (c) \(\text{Ann} (A_{\alpha\beta}+I_\beta)=I_\alpha\).

En route to proving Criterion A, a characterization is given using the Loewy factors of when a right semiartinian ring is regular with each primitive factor artinian. This produces a sequence of invariants for such rings from which the bounded nilpotency index is easily calculated and leads to a criterion of steadiness of a subclass of these rings called almost abelian regular rings. Examples are given to illustrate the difficulties in obtaining further criteria. The paper ends with a short section on steadiness vis à vis pure extensions.

For the entire collection see [Zbl 0897.00025].

The class of dually slender modules is occasionally much larger than that of the finitely generated ones and so it is useful to consider the following intermediate classes of modules. For any infinite cardinal \(\kappa\), a module \(M\) is called \(\kappa\)-reducing if each \(\kappa\)-generated submodule of \(M\) is contained in a finitely generated one. Then for all cardinals \(\omega<\kappa<\lambda\) we have finitely generated \(\Rightarrow\lambda\)-reducing \(\Rightarrow\kappa\)-reducing \(\Rightarrow\omega\)-reducing \(\Rightarrow\) dually slender. Eklof, Goodearl and Trlifaj (loc. cit.) have shown that, in general, these implications are irreversible. (Note that \(M\) is finitely generated iff it is \(\kappa\)-reducing for all infinite \(\kappa\).)

The aim of the paper under review is to provide a ring-theoretic criterion of steadiness for particular classes of rings and a general characterization of when all \(\omega\)-reducing modules are finitely generated. We now record the two main results, Criteria A and B.

Criterion A says that if \(R\) is a commutative semiartinian ring then \(R\) is not steady iff there is a two-sided ideal \(J\) of \(R\) containing the Jacobson radical of \(R\) and a member \(I\) of the Loewy (socle) series of the ring \(R/J\) such that \(I\) is an infinitely generated dually slender right \(R/J\)-module. Criterion B says that, letting \(M_n(R)\) denote the ring of \(n\times n\) matrices over the ring \(R\), there is an infinitely generated \(\omega\)-reducing \(R\)-module iff for some finite \(n\) there is a family \(\{A_{\alpha\beta}:\alpha<\beta<\omega_1\}\) of elements of \(M_n(R)\) and a family \(\{I_\alpha:\alpha<\omega_1\}\) of right ideals of \(M_n(R)\) such that, for all \(\alpha<\beta<\gamma<\omega_1\), (a) \(A_{\beta\gamma} A_{\alpha\beta}+I_\gamma=A_{\alpha\gamma}+I_\gamma\), (b) for all \(A\in M_n(R)\), \(A_{\alpha\gamma}A+ I_\gamma\neq A_{\beta\gamma}+ I_\gamma\) and (c) \(\text{Ann} (A_{\alpha\beta}+I_\beta)=I_\alpha\).

En route to proving Criterion A, a characterization is given using the Loewy factors of when a right semiartinian ring is regular with each primitive factor artinian. This produces a sequence of invariants for such rings from which the bounded nilpotency index is easily calculated and leads to a criterion of steadiness of a subclass of these rings called almost abelian regular rings. Examples are given to illustrate the difficulties in obtaining further criteria. The paper ends with a short section on steadiness vis à vis pure extensions.

For the entire collection see [Zbl 0897.00025].

Reviewer: J.Clark (Dunedin)

##### MSC:

16D80 | Other classes of modules and ideals in associative algebras |

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

16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |

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