Preliminaries.- Basic Concepts of Difference Algebra.- Difference Modules.- Difference Field Extensions.- Compatibility, Replicability, and Monadicity.- Difference Kernels over Partial Difference Fields. Difference Valuation Rings.- Systems of Algebraic Difference Equations.- Elements of the Difference Galois Theory.

Difference algebra grew out of the study of algebraic difference equations with coefficients from functional fields in much the same way as the classical algebraic geometry arose from the study of polynomial equations with numerical coefficients. The first stage of the development of the theory is associated with its founder J. F. Ritt (1893 - 1951) and R. Cohn whose book Difference Algebra (1965) remained the only fundamental monograph on the subject for many years. Nowadays, difference algebra has overgrew the frame of the theory of ordinary algebraic difference equations and appears as a rich theory with applications to the study of equations in finite differences, functional equations, differential equations with delay, algebraic structures with operators, group and semigroup rings. This book reflects the contemporary level of difference algebra; it contains a systematic study of partial difference algebraic structures and their applications, as well as the coverage of the classical theory of ordinary difference rings and field extensions. The monograph is intended for graduate students and researchers in difference and differential algebra, commutative algebra, ring theory, and algebraic geometry. It will be also of interest to researchers in computer algebra, theory of difference equations and equations of mathematical physics. The book is self-contained; it requires no prerequisites other than knowledge of basic algebraic concepts and mathematical maturity of an advanced undergraduate.

First monograph on difference algebra that covers partial algebraic structures, and the only monograph on the subject published in the last forty years Contains new ideas and technique (such as construction of Gröbner bases with respect to several orderings and the concepts of multivariable dimension polynomials) that can be efficiently applied in various areas of algebra and algebraic geometry Contains an important application of the algebraic technique to the study of the A. Einstein's concept of strength of systems of difference equations of mathematical physics