Rings and factorization.

*(English)*Zbl 0674.13008
Cambridge: Cambridge University Press. ix, 111 p. £7.50/pbk; $ 15.95/pbk (1987).

The book under review is about rings and factorial rings with applications to problems in algebra and number theory. It arosed out of a course of 20 lectures to the second year mathematics students at the University of Sheffield.

With a minimum of prerequisites, the reader is introduced to the notions of unique factorization domains, Euclidean domains, factorization of polynomials. - The book contains numerous examples and many exercises with hints and solutions.

As the author says the aim of the above mentioned course, and consequently also the aim of the book, is to help students to make the transition into a more abstract world as painless as possible by presenting abstract ideas in a fairly concrete context. - It seems to the reviewer that his aim is obtained except, perhaps, in one point, where the author defines a polynomial and a formal power series over an arbitrary ring R as a “formal expression”: \(a_ 0+a_ 1x+a_ 2x^ 2+...,\) \(a_ i\in R,\) with x ‘variable’ or indeterminate. In this way there is confusion between \(``+''\) appearing in the formal expression and \(``+''\) denoting addition of polynomials. It is much better to define x as a transcendental element over R contained in a suitable overring \(R'\supset R\), by showing later that R determines in a natural way R[x]; the second year mathematics students need to know where x is and what x is, in particular to know that x is a well determined element in \(R'\).

With a minimum of prerequisites, the reader is introduced to the notions of unique factorization domains, Euclidean domains, factorization of polynomials. - The book contains numerous examples and many exercises with hints and solutions.

As the author says the aim of the above mentioned course, and consequently also the aim of the book, is to help students to make the transition into a more abstract world as painless as possible by presenting abstract ideas in a fairly concrete context. - It seems to the reviewer that his aim is obtained except, perhaps, in one point, where the author defines a polynomial and a formal power series over an arbitrary ring R as a “formal expression”: \(a_ 0+a_ 1x+a_ 2x^ 2+...,\) \(a_ i\in R,\) with x ‘variable’ or indeterminate. In this way there is confusion between \(``+''\) appearing in the formal expression and \(``+''\) denoting addition of polynomials. It is much better to define x as a transcendental element over R contained in a suitable overring \(R'\supset R\), by showing later that R determines in a natural way R[x]; the second year mathematics students need to know where x is and what x is, in particular to know that x is a well determined element in \(R'\).

Reviewer: E.Stagnaro

##### MSC:

13F15 | Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial) |

13A05 | Divisibility and factorizations in commutative rings |

13B25 | Polynomials over commutative rings |

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