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Goldie's Theorem implies that a semiprime left Goldie ring is embeddable intoa semisimple Artinian ring. On the other hand, there are domains that are notembeddable into division rings. A criterion for a semiprime ring beingembeddable into a semisimple Artinian ring is given. Three types of embeddingsof semiprime rings into semisimple Artinian rings are introduced and studied:the elementary, natural and collection embeddings. It is proven that for eachembedding there is an elementary embedding which is smaller and explicitlyconstructed. In particular, the minimal embeddings are elementary ones. It isproven that a semiprime ring and all its localizations at regular left Ore setshave the `same' sets of elementary embeddings. A new criterion for the leftquotient ring of a ring being a semisimple Artinian ring is given in terms ofthe centre of the ring which is completely different from Goldie's Theorem(which is historically the first criterion, \cite{Goldie-PLMS-1960}). It isproven that the minimal embeddings in each Morita equivalence class ofembeddings is a {\em non-empty} set and they are explicitly described.Descriptions of the minimal embeddings of semiprime left Goldie rings and alltheir localizations at regular left Ore sets are described.

Embeddings of semiprime rings into semisimple Artinian rings