Commutative coherent rings: Historical perspective and current developments. (English) Zbl 0787.13001

Summary: Coherent ring theory constitutes a major area of research in commutative algebra. The wealth of results in this area, beside being of interest in their own right, contributed to the development of the general theory of commutative rings and influences research in other areas of algebra.
This article surveys the evolution of the theory of coherent rings since its beginning in 1960. The introduction describes the origin of the notion of coherence in algebra and the amalgamation of several research trends in diverse disciplines such as algebraic geometry, complex analysis, number theory and computer science, which gave rise to the definition of a coherent ring. The main body of this article concentrates on the description of the most significant topics of research in coherent ring theory and includes for each topic a mathematical history, statements of the main results and their contribution to the development of the general theory of commutative rings, and a probing of possible future research direction. The article concludes by briefly highlighting the impact of the research in coherent ring theory on other areas of algebra.


13-03 History of commutative algebra
13E15 Commutative rings and modules of finite generation or presentation; number of generators