An introduction to noncommutative Noetherian rings.

*(English)*Zbl 0679.16001
London Mathematical Society Student Texts, 16. Cambridge ect.: Cambridge University Press. xvii, 303 p. £30.00/hbk; £11.95/pbk; $ 49.50/hbk; $ 19.95/pbk (1989).

In the setting of commutative ring theory, the noetherian condition arises very naturally as it is satisfied by the rings of algebraic integers arising in number theory and the coordinate rings crucial to algebraic geometry. Until the arrival of Goldie’s Theorem in 1958, however, the maximum condition for one-sided ideals played only a minor role in the noncommutative theory. Indeed, the last definitive account of pre-Goldie ring theory, N. Jacobson’s book “Structure of Rings” published in 1956 (Zbl 0073.020), barely mentions noetherian rings. The principal reason was that localization, one of the main tools of the commutative theory was lacking, and Malcev’s example of a noncommutative domain that did not have a skew field of fractions left for almost two decades considerable doubt as to whether a general quotient ring theorem could be formulated.

Thus, Goldie’s proof of the existence of a simple artinian ring of fractions for a noetherian prime ring marks the birth of the extensive theory of noncommutative noetherian rings as we know it today. Not only did this result give tremendous impetus to the abstract theory, but, being widely applicable to special classes of rings, it provided feedback to the general theory from such diverse areas as enveloping algebras of finite dimensional Lie algebras, rings with polynomial identity, and certain infinite group rings.

In now takes a monumental effort to provide a reasonably complete account of the proliferation of results and methods that has taken place during the past thirty years. One such project having recently been completed, and very successfully so, by J. C. McConnell and J. C. Robson in their book “Noncommutative Noetherian Rings” (1987; Zbl 0644.16008), the present authors aim at a more modest goal. Instead, they provide mainly an introduction to the subject, be that for the mature mathematician interested in learning about noetherian rings or for the graduate student looking for an area to eventually writing a thesis in. Thus, the treatment is definitely not encyclopedic and overloaded with technical details and miraculous arguments that work but once, but aims at presenting the main threads and current topics of the theory and emphasizes results and methods that can be used in a wide context. One of the main themes of the text is to present most rings as rings of transformations or operators, not only because noncommutative rings frequently arise as such, but also because this point of view is crucial in many applications. While the authors deliberately chose not to focus on such areas as polynomial identity rings, group algebras, differential operator rings, and enveloping algebras, each of these having an extensive theory of its own, they nevertheless sketch an introduction to these subjects by looking at surrogates - algebras that are finitely generated as modules over commutative rings for the first, skew-Laurent rings for the second, and formal differential operator rings for the last two. Thus, the reader still gets the flavour of each of these subjects which represent the major fields of application of the general theory, without being side-tracked and losing the general point of view.

In order to sketch the contents of the book, we give a list of the titles of each chapter. (1) A few noetherian rings. (2) Prime ideals. (3) Semisimple modules, artinian modules, and nonsingular modules. (4) Injective hulls. (5) Semisimple rings of fractions. (6) Modules over semiprime Goldie rings. (7) Bimodules and affiliated prime ideals. (8) Fully bounded rings. (9) Rings of fractions. (10) Artinian quotient rings. (11) Links between prime ideals. (12) Rings satisfying the second layer condition. (13) Krull dimension. (14) Numbers of generators of modules. (15) Transcendental division algebras. Some bibliographical notes are included at the end of each chapter which provide an idea of the historical sources of the theory and point the reader in the direction of specialized accounts. These are very useful and remarkably accurate, considering that it is often quite difficult to pinpoint the precise sources of ideas in a rapidly evolving theory. Also quite useful are the many exercises, most of which are woven into the text with the intent of giving the reader some practical experience and of testing his/her understanding as he/she moves along. The appendix presents twenty research problems, each with a sketch of its history and of partial results. These are questions that all have been resolved for commutative noetherian rings, many have negative answers for merely one-sided noetherian rings, and, according to the authors, these problems test whether the theory of a specific class of noetherian rings, e.g. rings satisfying the second layer condition, is sufficiently developed so that they can be resolved within that class. All in all a well written, attractive book, remarkably free of misprints, that will be useful for both expert and novice for years to come.

Thus, Goldie’s proof of the existence of a simple artinian ring of fractions for a noetherian prime ring marks the birth of the extensive theory of noncommutative noetherian rings as we know it today. Not only did this result give tremendous impetus to the abstract theory, but, being widely applicable to special classes of rings, it provided feedback to the general theory from such diverse areas as enveloping algebras of finite dimensional Lie algebras, rings with polynomial identity, and certain infinite group rings.

In now takes a monumental effort to provide a reasonably complete account of the proliferation of results and methods that has taken place during the past thirty years. One such project having recently been completed, and very successfully so, by J. C. McConnell and J. C. Robson in their book “Noncommutative Noetherian Rings” (1987; Zbl 0644.16008), the present authors aim at a more modest goal. Instead, they provide mainly an introduction to the subject, be that for the mature mathematician interested in learning about noetherian rings or for the graduate student looking for an area to eventually writing a thesis in. Thus, the treatment is definitely not encyclopedic and overloaded with technical details and miraculous arguments that work but once, but aims at presenting the main threads and current topics of the theory and emphasizes results and methods that can be used in a wide context. One of the main themes of the text is to present most rings as rings of transformations or operators, not only because noncommutative rings frequently arise as such, but also because this point of view is crucial in many applications. While the authors deliberately chose not to focus on such areas as polynomial identity rings, group algebras, differential operator rings, and enveloping algebras, each of these having an extensive theory of its own, they nevertheless sketch an introduction to these subjects by looking at surrogates - algebras that are finitely generated as modules over commutative rings for the first, skew-Laurent rings for the second, and formal differential operator rings for the last two. Thus, the reader still gets the flavour of each of these subjects which represent the major fields of application of the general theory, without being side-tracked and losing the general point of view.

In order to sketch the contents of the book, we give a list of the titles of each chapter. (1) A few noetherian rings. (2) Prime ideals. (3) Semisimple modules, artinian modules, and nonsingular modules. (4) Injective hulls. (5) Semisimple rings of fractions. (6) Modules over semiprime Goldie rings. (7) Bimodules and affiliated prime ideals. (8) Fully bounded rings. (9) Rings of fractions. (10) Artinian quotient rings. (11) Links between prime ideals. (12) Rings satisfying the second layer condition. (13) Krull dimension. (14) Numbers of generators of modules. (15) Transcendental division algebras. Some bibliographical notes are included at the end of each chapter which provide an idea of the historical sources of the theory and point the reader in the direction of specialized accounts. These are very useful and remarkably accurate, considering that it is often quite difficult to pinpoint the precise sources of ideas in a rapidly evolving theory. Also quite useful are the many exercises, most of which are woven into the text with the intent of giving the reader some practical experience and of testing his/her understanding as he/she moves along. The appendix presents twenty research problems, each with a sketch of its history and of partial results. These are questions that all have been resolved for commutative noetherian rings, many have negative answers for merely one-sided noetherian rings, and, according to the authors, these problems test whether the theory of a specific class of noetherian rings, e.g. rings satisfying the second layer condition, is sufficiently developed so that they can be resolved within that class. All in all a well written, attractive book, remarkably free of misprints, that will be useful for both expert and novice for years to come.

Reviewer: G.Krause

##### MSC:

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

16P40 | Noetherian rings and modules (associative rings and algebras) |

16P50 | Localization and associative Noetherian rings |

16N60 | Prime and semiprime associative rings |

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

16Dxx | Modules, bimodules and ideals in associative algebras |