Equivalence relations, groups, subgroups, isomorphism of groups, transformation groups, realization of a group as a transformation group, cyclic groups, order of an element, invariant subgroups, factor groups, homomorphism of groups, the fundamental theorem of homomorphism for groups, abelian groups, classification of finitely generated abelian groups; linear representations of groups, irreducible representations, characters of representations; rings, commutative rings, integral domains, division rings, quarternions, fields, matrix rings, polynomial rings, subrings, center of a ring, ideals, quotient rings, ideals and quotient rings for the ring of integers, homomorphism of rings, fundamental theorem of homomorphisms of rings, simple ideals, prime ideals, principal ideals, coordinate ring of an algebraic curve, Hilbert basis theorem, the Hilbert Nullstellensatz; fields, vector spaces over a field, field of fractions of a commutative integral domain, extensions of fields, simple extensions, subfields, prime fields, the structure of finite fields, the field of rational functions on an algebraic curve; Galois theory; basics of homological algebra and category theory: categories, functors, commutative diagrams, modules, projective and injective modules.