Non-algebraists frequently imagine an algebra textbook as a theorem-laden tome bare of concrete computations or applications to science or engineering. Over the past three or four decades, however, a community of algebraists has used computers to analyze and solve difficult problems and apply them in those settings. Researchers within this community have developed and refined software systems that allow them to revisit a number of questions that previously had been considered intractable due to their sheer computational burden. These systems have grown to the point where the last decade has seen the publication of a number of textbooks that use a computer algebra system as an integral tool for teaching commutative algebra and algebraic geometry. Among these is the text A SINGULAR Introduction to Commutative Algebra, recently released in a second edition. (Readers may also be interested in a review of the first edition of this text by John B. Little, in the March 2004 American Mathematical Monthly.)
Greuel and Pfister were two of the original developers of SINGULAR, a free, open-source, award-winning system for computing with rings and ideals; hence the title of this book. The textbook relies heavily on this software and includes it in an attached CD-ROM.
This is, however, a textbook on commutative algebra, not on a computer algebra system; the reader dives directly into rings from page one. SINGULAR is usually used to illustrate a concept explained in the text, or to explore a concept further. Detailed explanations of the various SINGULAR commands do not usually appear within the text, but in an appendix, so the rationale for some constructions might not be immediately apparent. Nevertheless, the language of SINGULAR is clear, and the examples sufficiently focused, that the reader can follow the examples without difficulty.
Books with a computational flavor mix constructive approaches, in the form of algorithms and computations, with theorems and proofs. Constructive approaches often reveal information hidden by elegant but abstract proofs; the authors of this text cite primary decomposition as a case where the constructive approach reveals a great deal more .
In most textbooks, the computational flavor manifests itself in the form of algorithms written in pseudocode and applications to areas such as robotics or financial mathematics. The general nature of pseudocode means that the reader can program the algorithm on any computer algebra system, so long as she knows how to translate pseudocode into that system's language. The drawback to this is that it can steepen the learning curve for students unaccustomed to programming. The focus on SINGULAR allows the authors to state precisely how the user should interact with the program and to provide procedures — not only pseudocode! — implementing certain algorithms. The authors remark that the procedures they provide are simplifications of SINGULAR library code.
One milestone in this field is the development by Bruno Buchberger of an algorithm to compute Gröbner bases in 1965. The theory of Gröbner bases allows one to resolve with relative ease many important questions regarding ideals of polynomial rings. These objects are fundamental to commutative algebra, algebraic geometry, and their applications, so that a discussion Gröbner bases and related methods is increasingly common in textbooks on these topics.
A distinguishing hallmark of A SINGULAR Introduction to Commutative Algebra is its focus on standard bases, which generalize the ideas of Gröbner basis theory to settings beyond polynomial rings. This naturally requires a generalization of many ideas fundamental to Gröbner bases, such as monomial orderings and normal forms.
The reader encounters right away (in Section 1.2) a distinction between global, local, and mixed monomial orderings. Most readers would find global monomial orderings natural: for any indeterminate x, 1 < x < x2 <…, and typical textbooks do not venture beyond this. A local ordering, by contrast, reverses this property: for any indeterminate x, 1 > x > x2 >…. Local orderings are necessary for standard bases of ideals of power series rings and local rings, useful for the analysis of singularities.
After a lengthy introduction to the theory of standard bases for rings and modules, the textbook treats a number of important topics in commutative algebra: Noether normalization, primary decomposition, the Hilbert function, and homological algebra. The three substantial appendices account for one-third of the book's nearly 700 pages, and include a foray into ideas of algebraic geometry related to topics in the text, a manual on SINGULAR, and a chapter on polynomial factorization.
This second edition of the text adds the previously-mentioned chapter on polynomial factorization, a section on non-commutative Gröbner bases written by Viktor Levandovskyy, and sections on characteristic sets and triangular sets. Unlike many texts in this field, it does not venture into applications that lie outside commutative algebra, although the appendix on SINGULAR gives a brief description of some SINGULAR libraries that are useful for visualization, coding theory, and system and control theory.
The writing is clear and precise. Most of the examples are excellent and insightful, although a few are a little trivial. A substantial number of exercises accompany each section; these range from using SINGULAR to verify a computation to proving some difficult theorems. The result should serve well as a graduate-level textbook in commutative algebra or as a resource for independent study.
The user should note that SINGULAR does not come with a worksheet interface like Maple or Mathematica, but has a terminal interface that operates within a shell.
John Perry is an assistant professor of mathematics at the University of Southern Mississippi. His mathematical interests lie primarily in computational algebra. He once had a number interests outside of mathematics, but after three children and several home renovations he has forgotten what it means to have free time.