Derivatives, nuclei and dimensions on the frame of torsion theories.

*(English)*Zbl 0661.16020
Pitman Research Notes in Mathematics Series, 188. Harlow (UK): Longman Scientific & Technical; New York: John Wiley & Sons. 120 p. £13.50 (1988).

Given a ring R, the family R-tors of all hereditary torsion theories over R-mod forms a lattice called a frame, i.e. a complete lattice in which meets distribute over infinite joins. This gives us the possibility of analyzing R-tors using topological techniques and the more lattice- theoretic techniques developed in the frame theory. In this monograph the authors concentrate on the study of the set of derivatives, prenuclei and nuclei on R-tors which are known to be important tools in topology and general frame theory and are also relevant to module theory. The general theory of these objects is developed by H. Simmons.

The monograph is divided into five chapters: Chapter 0, Background material; Chapter 1, Derivatives on R-tors; Chapter 2, Nuclei on R-tors; Chapter 3, Some further constructions; Chapter 4, Dimensions defined by chains of derivatives.

Chapter 0 starts with the definition of the topologizing filter of left ideals of R and then the frame of torsion theories on R-mod. A derivative on R-tors is a function z from R-tors to itself satisfying the following conditions: (1) \(\tau\leq z(\tau)\) for all \(\tau\in R\)-tors; (2) If \(\sigma\leq \tau\) in R-tors then \(z(\sigma)\leq z(\tau).\)

In Chapter 1 several examples of specific derivatives are given. The set of all derivatives on R-tors has the structure of a complete lattice, a semigroup and also an ordered monoid. These are discussed in this chapter. A prenucleus on R-tors is a derivative z on R-tors satisfying (3) \(z(\sigma \wedge \tau)=z(\sigma)\wedge z(\tau)\) for all \(\sigma,\tau\in R\)-tors and a nucleus on R-tors is an idempotent prenucleus.

Chapter 2 begins with a collection of examples of nuclei. A way in which nuclei arise naturally on R-tors is obtained from the general radical theory on complete lattices as defined by S. A. Amitsur and the authors point out that every nucleus on R-tors arises in this manner.

In Chapter 3 the notions of the radical of prenuclei and the fixed set of a derivative are introduced. It is shown that an analog of the notion of Jacobson radical of a ring holds and that a necessary and sufficient condition for a derivative to be a nucleus in terms of the fixed set of a derivative.

In Chapter 4, the authors indicate how chains of derivatives can be used to define the general notion of dimensions on the module category and how some of the dimensions used in the module category, such as Gabriel dimensions and TTK-dimensions, arise in this manner.

In the preface of the monograph, the authors tell us that there is still much to be discovered and they hope that after reading the monograph the reader will be in a better position to contribute to this search.

The monograph is divided into five chapters: Chapter 0, Background material; Chapter 1, Derivatives on R-tors; Chapter 2, Nuclei on R-tors; Chapter 3, Some further constructions; Chapter 4, Dimensions defined by chains of derivatives.

Chapter 0 starts with the definition of the topologizing filter of left ideals of R and then the frame of torsion theories on R-mod. A derivative on R-tors is a function z from R-tors to itself satisfying the following conditions: (1) \(\tau\leq z(\tau)\) for all \(\tau\in R\)-tors; (2) If \(\sigma\leq \tau\) in R-tors then \(z(\sigma)\leq z(\tau).\)

In Chapter 1 several examples of specific derivatives are given. The set of all derivatives on R-tors has the structure of a complete lattice, a semigroup and also an ordered monoid. These are discussed in this chapter. A prenucleus on R-tors is a derivative z on R-tors satisfying (3) \(z(\sigma \wedge \tau)=z(\sigma)\wedge z(\tau)\) for all \(\sigma,\tau\in R\)-tors and a nucleus on R-tors is an idempotent prenucleus.

Chapter 2 begins with a collection of examples of nuclei. A way in which nuclei arise naturally on R-tors is obtained from the general radical theory on complete lattices as defined by S. A. Amitsur and the authors point out that every nucleus on R-tors arises in this manner.

In Chapter 3 the notions of the radical of prenuclei and the fixed set of a derivative are introduced. It is shown that an analog of the notion of Jacobson radical of a ring holds and that a necessary and sufficient condition for a derivative to be a nucleus in terms of the fixed set of a derivative.

In Chapter 4, the authors indicate how chains of derivatives can be used to define the general notion of dimensions on the module category and how some of the dimensions used in the module category, such as Gabriel dimensions and TTK-dimensions, arise in this manner.

In the preface of the monograph, the authors tell us that there is still much to be discovered and they hope that after reading the monograph the reader will be in a better position to contribute to this search.

Reviewer: Y.Kurata

##### MSC:

16S90 | Torsion theories; radicals on module categories (associative algebraic aspects) |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

18E40 | Torsion theories, radicals |

16Nxx | Radicals and radical properties of associative rings |

16P60 | Chain conditions on annihilators and summands: Goldie-type conditions |

06B23 | Complete lattices, completions |