Elementary Commutative Algebra

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.