Modules over PID with applications to the structure of finitely generated abelian groups and canonical forms of matrices. Basic theory of commulative rings: Localization, integral dependence, Noetherian and Artinian rings, Hilbert's Nullstellensatz, Hilbert series of graded algebras with applications to simplicial complexes, lattice points of convex polytopes and counting magic squares.
Texts / References:
N. Jacobson, Basic Algebra, Vol. 1 and 2, Hindustan Publishing Corporation, 1984.
S. Lang, Algebra, Revised 3rd ed., Springer (India), 2004.