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Recent advances in non-linear elliptic and parabolic problems , edited by P. Bénilan, M. Chipot, L. C. Evans and M. Pierre. Pp 340. £25. 1989. ISBN 0-582-03538-4 (Longman)

The first of these two volumes is (with minor changes) a translation of Chapters 1-4 of the authors' 1985 Polish text Commutative rings, and the second derives similarly from Chapters 5-8 of that work. Although there are many books on this subject in English, the original Polish book differs in choice of material and its presentation, emphasising connections with algebraic geometry and algebraic number-theory. The introductory Chapter 1 of the first volume under consideration was originally intended to serve the whole book; thus some of its material is now not required until the second volume, but this format was preserved "to make both parts, treated as a whole, as selfcontained as possible". (Some indication of topics which could be deferred would have been helpful.) This chapter (42 pages), entitled Preliminary concepts, presents the fundamentals concerning rings, ideals and modules; the basic relationships between algebraic geometry and commutative ring theory; and results on localisation and on graded rings and modules. The level and tone of the volume can be gauged from this list (almost a course in itself!). Titles and lengths of the remaining chapters are: Noetherian rings and modules (51); Integral domains and Dedekind domains (49); Divisors and Krull domains (52). Although categorical and homological concepts appear throughout the first volume, the second is specifically concerned with applications of homological algebra to ring theory. (It was the successful use of homological methods in the 1950s to answer a long-standing question about regular local rings that drew attention to these powerful ideas, which then rapidly permeated many other parts of mathematics.) An appendix (22 pages) summarises the quite considerable background knowledge required, and is supported by references to readily accessible texts in English. The chapters are: Dimension and multiplicity (77); Regular local rings (22); Cohen-Macaulay rings (36); Gorenstein rings (20). There is a convenient list of all definitions and results quoted from the first volume. In both books almost all the chapter sections include exercises (the harder of which have hints), although exercise material is seldom used in the text. The chapters end with interesting "notes and references" of a historical nature. Each volume has a bibliography and index. The translation reads smoothly, and appears to have been carefully edited. Well printed in Poland on somewhat coarse paper, the volumes are colourfully bound. The authors have been careful to motivate the study of their subject by indicating its interactions with other parts of mathematics, but these readable, scholarly and costly books are really for the serious and well-prepared (postgraduate?) student of commutative algebra.