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This book studies representable functors among well-known varieties of algebras. All such functors from associative rings over a fixed ring $R$ to each of the categories of abelian groups, associative rings, Lie rings, and to several others are determined. Results are also obtained on representable functors on varieties of groups, semigroups, commutative rings, and Lie algebras. The book includes a 'Symbol index', which serves as a glossary of symbols used and a list of the pages where the topics so symbolized are treated, and a 'Word and phrase index'. The authors have strived - and succeeded - in creating a volume that is very user-friendly.