This book, a sequel to the authors’ previous

*Group Theory*, is the second in a projected series of algebra textbooks. It has much in common with its predecessor. Like

*Group Theory*, this book is based on lectures given in India to second and third year undergraduate students. Also like

*Group Theory*, the discussion of rings here is elementary: of the seven chapters in the text, six cover material generally covered in a first-semester “rings first” course in abstract algebra: the definition of rings, integral domains, and fields; homomorphisms and isomorphisms; ideals and quotient rings; polynomials and factorization. Chapter 7 introduces the general concept of divisibility in integral domains and discusses unique factorization domains, principal ideal domains, and Euclidean domains.

The word “group” is used many times in this book without definition, and some basic facts about groups (for example, Lagrange’s theorem) are used in the book on occasion, so perhaps some prior exposure to group theory would be valuable for a person reading this book.

Field theory is not developed at any great length in the book, although the authors do use quotient rings (they call them factor rings) to construct fields of prime-power order. However, concepts like separable and normal extensions, splitting fields, and so on, are not reached; neither, of course, is Galois theory.

This book incorporates the good features of its predecessor. The writing is clear and detailed. Most everything is proved in the text, although there are a few exceptions when appropriate; for example, although the authors give an example of a PID that is not a Euclidean domain, they do not prove that fact (though they do give a reference for a proof.) Examples are used extensively. (I particularly liked the use of quadratic domains.) Solved problems are interspersed, as appropriate, throughout the book. Exercises are plentiful, some but not all of them with hints. These features enhance the value of this book for self-study.

Unfortunately, however, some of the not-so-good features of *Group Theory* appear here as well. The authors’ treatment of the rings \( Z_{n} \) for example, is problematic. The set \( Z_{n} \) is defined as the set of integers \( \left\{0, 1, \ldots, n-1 \right\} \) with addition and multiplication defined as the remainders when the integers \( a + b \) and \( ab \) are divided by \( n \). In *Group Theory* the authors gave short shrift to the question of showing associativity of these operations; here, with the added complication of showing distributivity, the authors do not even make a pretense of offering a proof; they just state, without proof (or any indication that a proof is needed), that \( Z_{n} \) is a ring.

Also, the authors’ writing style can be idiosyncratic at times. Definite articles are frequently missing (“Unity is also a unit as unity is its own inverse.”; “The decomposition of integer into product of powers of primes…”; “Since characteristic of a field is either zero or prime…”) or misused (a “ring may or may not have the identity element with respect to multiplication.”). Other words are also sometimes used incorrectly; Liouville, for example, is referred to as a “French contribution” to mathematics rather than a “French contributor”. Some results seem garbled: one solved exercise, for example, asks the reader to prove that the characteristic of a finite integral domain is finite, even though the reader already knows that the characteristic of an integral domain is zero or a prime. The authors prove this by showing that the characteristic of the ring can’t be zero and then conclude the proof by noting that a prime \( p \) is finite. But of course zero is finite too, so all of this seems like a lot of verbiage that accomplishes nothing and may confuse the reader.

Also, as in the first book, it may be hard to find an audience for this one. It doesn’t seem likely that many American mathematics departments offer a course devoted entirely to ring theory. And, given that there are a number of general abstract algebra books at the undergraduate level that are also quite clear and detailed, and cover a much more varied selection of material, it seems unclear what particular advantage is to be gained by using as a text a book with a much narrower focus. But, as noted above, this book may find some use as a text for self-study.