Embeddings of semiprime rings into semisimple Artinian rings

Abstract

Goldie's Theorem implies that a semiprime left Goldie ring is embeddable into a semisimple Artinian ring. On the other hand, there are domains that are not embeddable into division rings. A criterion for a semiprime ring being embeddable into a semisimple Artinian ring is given. Three types of embeddings of semiprime rings into semisimple Artinian rings are introduced and studied: the elementary, natural and collection embeddings. It is proven that for each embedding there is an elementary embedding which is smaller and explicitly constructed. In particular, the minimal embeddings are elementary ones. It is proven that a semiprime ring and all its localizations at regular left Ore sets have the `same' sets of elementary embeddings. A new criterion for the left quotient ring of a ring being a semisimple Artinian ring is given in terms of the centre of the ring which is completely different from Goldie's Theorem (which is historically the first criterion, \cite{Goldie-PLMS-1960}). It is proven that the minimal embeddings in each Morita equivalence class of embeddings is a {\em non-empty} set and they are explicitly described. Descriptions of the minimal embeddings of semiprime left Goldie rings and all their localizations at regular left Ore sets are described.