##
**Topological rings.**
*(English)*
Zbl 0785.13008

North-Holland Mathematics Studies. 178. Amsterdam: North-Holland. x, 498 p. (1993).

This book is addressed both for beginners and for mature researchers in the field of topological rings and modules. The chapters I–IV can be viewed as an introduction in the theory of topological rings and modules.

In chapter I the notions of topological module and topological ring are introduced and elementary properties of their topologies are discussed. Examples of topological rings are indicated. Between them normed rings and rings with absolute values are presented. Here characteristic properties of neighborhoods of zero of topological rings and modules are given. The author proves elementary properties of subrings, ideals, quotients and projective limits of topological rings.

In chapter II a criterion of metrizability of a topological abelian group is given. Moreover, theorems on completions of topological abelian groups and rings are proved.

In the first part of chapter III the author describes elementary properties of bounded subsets of topological rings and modules. Further, he indicates some sufficient conditions for the topology of a topological ring to be given by a norm. Relations between norms and absolute values on fields are established. – In sections 15 and 16 properties of topological vector spaces over topological division rings are studied. The uniqueness of topology of finite-dimensional vector spaces over some topological division rings is proved, the finite-dimensionality of locally compact vector spaces over nondiscrete locally compact division rings is proved. In those sections some well-known theorems (theorem on the structure of connected locally compact division rings and theorem on normed division \(\mathbb{C}\)-algebras) are contained.

Chapter IV contains some results on real valuations on a ring, valuation rings, discrete valuations of division rings. In this chapter a proof is given of the fact that topology of a nondiscrete locally compact division ring can be induced by an absolute value.

In chapter V methods and notions of commutative algebra are used. Rings studied here are often furnished with \(I\)-adic topology where \(I\) is an ideal of a ring. Here the results on the existence of Cohen subrings in complete local rings are presented. In this chapter characterization of all complete discretely valued fields is given. It is established that every complete semilocal noetherian ring is the topological direct sum of finitely many complete local noetherian rings.

This theme is continued in chapter IX. Moreover, other topics on commutative algebra are discussed in this chapter, in particular, complete regular rings and the japanese property of complete Cohen noetherian domains.

The most results of chapter VI are purely algebraic. It contains results concerning primitive rings, the Jacobson radical of a ring, artinian modules and rings.

Chapters VII, VIII, X are devoted to the study of linearly compact rings and their modifications. In chapter VII the author gives the proof of a theorem concerning the structure of linearly compact semisimple rings. Here classical results about the structure of compact and locally compact semisimple rings are presented. In particular, a theorem on the structure of left bounded locally compact semisimple rings is proved.

In chapter VIII linearly compact rings with a nontrivial Jacobson radical are studied. Here results about the lifting of idempotents in linearly compact rings, about the summability of families of orthogonal idempotents in linearly compact rings are proved. In this chapter the study of properties of locally compact rings is continued. In particular, the properties of the connected component of a locally compact ring are proved. Some facts concerning simple locally compact rings are given. Further, here linearly compact and compact rings whose topology is given by the powers of the Jacobson radical are discussed.

In chapter X locally strictly linearly compact and locally centrally linearly compact rings are studied. The classification of finite- dimensional indecomposable Hausdorff algebras with identity over complete, discretely valued fields is given.

Chapter XI is a historical survey of the theory of topological rings.

Each section contains a list of exercises. The book has a quite complete bibliography on topological rings and modules.

In chapter I the notions of topological module and topological ring are introduced and elementary properties of their topologies are discussed. Examples of topological rings are indicated. Between them normed rings and rings with absolute values are presented. Here characteristic properties of neighborhoods of zero of topological rings and modules are given. The author proves elementary properties of subrings, ideals, quotients and projective limits of topological rings.

In chapter II a criterion of metrizability of a topological abelian group is given. Moreover, theorems on completions of topological abelian groups and rings are proved.

In the first part of chapter III the author describes elementary properties of bounded subsets of topological rings and modules. Further, he indicates some sufficient conditions for the topology of a topological ring to be given by a norm. Relations between norms and absolute values on fields are established. – In sections 15 and 16 properties of topological vector spaces over topological division rings are studied. The uniqueness of topology of finite-dimensional vector spaces over some topological division rings is proved, the finite-dimensionality of locally compact vector spaces over nondiscrete locally compact division rings is proved. In those sections some well-known theorems (theorem on the structure of connected locally compact division rings and theorem on normed division \(\mathbb{C}\)-algebras) are contained.

Chapter IV contains some results on real valuations on a ring, valuation rings, discrete valuations of division rings. In this chapter a proof is given of the fact that topology of a nondiscrete locally compact division ring can be induced by an absolute value.

In chapter V methods and notions of commutative algebra are used. Rings studied here are often furnished with \(I\)-adic topology where \(I\) is an ideal of a ring. Here the results on the existence of Cohen subrings in complete local rings are presented. In this chapter characterization of all complete discretely valued fields is given. It is established that every complete semilocal noetherian ring is the topological direct sum of finitely many complete local noetherian rings.

This theme is continued in chapter IX. Moreover, other topics on commutative algebra are discussed in this chapter, in particular, complete regular rings and the japanese property of complete Cohen noetherian domains.

The most results of chapter VI are purely algebraic. It contains results concerning primitive rings, the Jacobson radical of a ring, artinian modules and rings.

Chapters VII, VIII, X are devoted to the study of linearly compact rings and their modifications. In chapter VII the author gives the proof of a theorem concerning the structure of linearly compact semisimple rings. Here classical results about the structure of compact and locally compact semisimple rings are presented. In particular, a theorem on the structure of left bounded locally compact semisimple rings is proved.

In chapter VIII linearly compact rings with a nontrivial Jacobson radical are studied. Here results about the lifting of idempotents in linearly compact rings, about the summability of families of orthogonal idempotents in linearly compact rings are proved. In this chapter the study of properties of locally compact rings is continued. In particular, the properties of the connected component of a locally compact ring are proved. Some facts concerning simple locally compact rings are given. Further, here linearly compact and compact rings whose topology is given by the powers of the Jacobson radical are discussed.

In chapter X locally strictly linearly compact and locally centrally linearly compact rings are studied. The classification of finite- dimensional indecomposable Hausdorff algebras with identity over complete, discretely valued fields is given.

Chapter XI is a historical survey of the theory of topological rings.

Each section contains a list of exercises. The book has a quite complete bibliography on topological rings and modules.

Reviewer: V.Arnautov and M.Ursul (Kishinev)

### MSC:

13Jxx | Topological rings and modules |

16W80 | Topological and ordered rings and modules |

13-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra |