This module is designed for graduate students in mathematics. It covers the following major topics: Review of linear algebra: linear maps, determinants, duality, bilinear forms. Commutative rings and modules: projective and injective modules, tensor products, chain conditions, primary decomposition, Noetherian rings and modules, ring extensions, Dedekind domains. The structure of rings: primitive rings, the Jacobson radical, semisimple rings, division algebras. Homological algebra: complexes, homology sequence, Euler characteristic and the Grothendieck group, homotopies of morphisms of complexes.