"Ring, subring, ideal, operations on ideals, zero divisor elements, nilpotent elements, invertible elements, integral domain, field, prime ideal, maximal ideal, homomorphism of rings, quotient ring (with partitions and equivalence relations), isomorphism Theorem, The Chinese remainder Theorem, partially ordered sets, Zorn’slemma and existence of maximal ideals, nilradical, Jacobson ideal, localization of rings, Polynomial ring, Power series ring. Introduction to Noetherian rings: Module, submodule, operations on submodules, homomorphism of modules, finitely generated module, Noetherian modules, Noetherian rings, Hilbert’s Basis Theorem, Cohen’s criterion, primary decomposition. Introduction to UFD (unique factorization domain): Euclidean domain, PID (principal ideal domain), associate elements, irreducible element, prime element, connections between prime and irreducible, unique factorization domain, Hierarchy among UFD, PID, and Euclidean domains. Polynomial ring over UFD, Eisenstein’s Criterion. Introduction to integral dependence and valuations: Integral elements, integrally closed domain, going up, going down, totally ordered group, valuation, valuation domain, Prüfer domain, fractional ideal, invertible ideal, Dedekin domain."