The Handbook of Algebra is the third volume devoted to the contemporary algebra. The content of the algebraic research is divided into nine sections, namely: 1. Linear algebra, fields, algebraic number theory. 2. Category theory, homological and homotopical algebra, methods from logic (algebraic model theory). 3. Commutative and associative rings and algebras. 4. Other algebraic structures, non-associative rings and algebras, commutative and associative rings and algebras with extra structure. 5. Groups and semigroups. 6. Representations and invariant theory. 7. Machine computation, algorithms, tables. 8. Applied algebra. 9. History of algebra. The current volume deals with the first five sections. From the first section, the linear algebra over commutative rings is presented. As for the second section, the category theory, algebraic K-theory and model theoretic algebra are treated. Some parts from commutative and associative rings and algebras and coalgebras belonging to the third section are investigated. The last two parts of the book are devoted to the lattices and partially ordered sets, varietes of algebras and groups, Lie algebras and algebras with additional structures and groups and semigroups. It should be emphasized that at the beginning of the book, the outline of all three series of this Handbook is presented in details.