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The primary audience for this book is students and the young researchers interested in the core of the discipline. Commutative algebra is by and large a self-contained discipline, which makes it quite dry for the beginner with a basic training in elementary algebra and calculus. A stable mathematical discipline such as this enshrines a vital number of topics to be learned at an early stage, more or less universally accepted and practiced. Naturally, authors tend to turn these topics into an increasingly short and elegant list of basic facts of the theory. So, the shorter the better. However, there is a subtle watershed between elegance and usefulness, especially if the target is the beginner. From my experience throughout years of teaching, elegance and terseness do not do it, except much later in the carrier. To become useful, the material ought to carry quite a bit of motivation through justification and usefulness pointers. On the other hand, it is difficult to contemplate these teaching devices in the writing of a short book. I have divided the material in three parts. starting with more elementary sections, then carrying an intermezzo on more difficult themes to make up for a smooth crescendo with additional tools and, finally, the more advanced part, versing on a reasonable chunk of present-day steering of commutative algebra. Historic notes at the end of each chapter provide insight into the original sources and background information on a particular subject or theorem. Exercises are provided and propose problems that apply the theory to solve concrete questions (yes, with concrete polynomials, and so forth).

The foremost goal of the book is to provide an introductory text in commutative algebra that assumes a general knowledge of algebraic structures (rings and fields mainly) with sufficient maturity to handle residual structures and substructures, generators and homomorphisms. But no solid knowledge of group theory or field theory is needed, though some previous exposition to these topics may help build up that maturity. A second intent is to give a tinge of the modern ideas pervasive in the area, if not of its recent sophisticated methods. To help accomplishing it, I included after each section a subsection called Discussion, in which a mix of previous ideas and new ones are (sometimes only perfunctorily) exposed. A third motivation is to emphasize the effective side of the theory right at the outset. Computability is a fact of (modern) mathematics. Increasingly efficient algorithms stressing the symbolic side of computation have been developed. The idea here is to call the novice's attention to the underlying computational content of a definition or of a theorem. The book is based on lectures delivered during various years at the Mathematics Departament of UFPE.