Definition of Rings, Examples including Polynomial Rings, Formal Power Series Rings, Matrix Rings and Group Rings. Commutative Rings, Integral Domain, Division Ring, Characteristics of an Integral domain.
Homomorphisms, kernel, Isomorphism, Ideals, Quotient Rings, Maximal Ideals, the Field of Quotients of an Integral Domain.
Euclidean Rings, Principal Ideal, Unit Element, Greatest Common Divisor, Prime Elements, Unique Factorization Theorem, The ring of Gaussian integers, Fermat’s Theorem. Polynomial Rings – F[x], Degree of a Polynomial, The Division Algorithm, Principal Ideal Ring, Irreducible Polynomial a principal ideal ring, Irreducible polynomial, Polynomial Rings over the Rational Field, Primitive Polynomials, The Content of a Polynomial, Integer Monic Polynomial, Gauss Lemma, Eisenstein Criterion.
Polynomial Rings over Commutative Rings. Unique Factorization Domain. Euclidean Domains, Principal Ideal Domains, Unique Factorization Domains, and their proper inclusions .