These notes prove the basic theorems in commutative algebra required for algebraic geometry and algebraic groups. They assume only a knowledge of the algebra usually taught in advanced undergraduate or first-year graduate courses. However, they are quite concise.

Rings and algebras; ideals; noetherian rings; unique factorization; integrality; rings of fractions; direct limits; tensor products; flatness; finitely generated projective modules; the Hilbert Nullstellsatz; the max spectrum of a ring; dimension theory for finitely generated k-algebras; primary decompositions; artinian rings; dimension theory for noetherian rings; regular local rings; connections with geometry.