Introduction

Abstract

Could it be that Ring Theory is the part of mathematics where chaos is most present? Researchers in this field will recognize an apparent state of disorder caused on one hand by the existence of a large variety of different kinds of rings evolving from various areas of applications, while on the other hand few methods of general applicability are available. There seems to be a pseudo-philosophical basis for the idea that dimension theory may be a general applicable method in all fields of mathematics. Indeed, the concept of “dimension” is fundamental in the logical interpretation of our observations of reality and it must be inherent in the structure of human thought. But how can one then avoid the basic contradiction in trying to solve specific problems by a general method? We have tried to do so by proposing dimension theory, not as a general method, but as a unifying concept allowing for the differentiation of specific dimension-notions prompted by the consideration of different problems. Geometric intuition about dimension of spaces may without much trouble be extended to abstract algebraic objects like vector spaces and algebras over fields, but, when applied to rings and modules, this intuition will fail on many occasions. Therefore one quickly learns to comply with the fact that different problems in Ring Theory require different notions of “dimension”. Recent developments have established that this diversification does not lead to: just an enlargement of the available body of abstract nonsense.