Introduction

It seems to be a formidable open problem in ring theory to find an acceptably inclusive theoretical framework for describing, and also classifying and thus unifying the wide variety of simple rings that are known.

In applications of ring theory, algebras over a field are so frequently encountered that they cannot be ignored. Some of our previous purely ring theoretic concepts can be extended to algebras over a field of scalars F. This amounts to assuming in addition that all our submodules, subrings, and right ideals are F-subspaces while module homomorphisms are F-linear. Not only is the material in Section 1 necessary equipment of every ring theorist, but it will allow us to ignore all of these additional technical complications later when we will deal with simple modules and simple algebras.

The second section shows that a simple algebra as well as every simple ring, with or without identity, is an algebra over a field called its centroid. As the name suggests, when the ring has an identity, the centroid is essentially the center. All of this holds for arbitrary nonassociative algebras, which is the natural framework for establishing these facts. It is only the second section which briefly touches upon non-associative rings. Aside from studying our main object – simple rings, section two gives the reader a brief glimpse of one of the last remaining unexplored important frontiers of ring theory – nonassociative rings. Through all of this it should be remembered that every theorem established for nonassociative rings holds a fortiori for associative ones.