The aim of the course is to introduce the main concepts of both noncommutative and commutative algebra and some methods of homological algebra.

Lectures

Oral Examination

The contents of the courses: Algebra 1, Algebra 2, Linear Algebra and Geometry 1.

Modules over a ring and operations on modules. Semisimple rings and modules; Wedderburn-Artin theorem and applications to representation theory of finite groups. Artinian and noetherian rings and modules. Localization of commutative rings and of modules over them. Primary decomposition of ideals. Dimension theory. Integral dependence and valuations; Dedekind domains. Spectrum of a commutative ring; affine algebraic sets, Noether's normalization lemma and Hilbert's Nullstellensatz. Injective, projective and flat modules; resolutions of modules. Abelian categories, left or right exact functors and derived functors; Ext and Tor functors; group cohomology

P. Aluffi: "Algebra: chapter 0", Graduate Studies in Mathematics 104, American Mathematical Society, 2009.
M.F. Atiyah, I.G. MacDonald: "Introduzione all'algebra commutativa", Feltrinelli, 1981.
S. Bosch: "Algebraic Geometry and Commutative Algebra", Universitext, Springer, 2013.
P.J. Hilton, U. Stammbach: "A Course in Homological Algebra", second edition, Graduate Texts in Mathematics 4, Springer-Verlag, 1997.
T.Y. Lam: "A first course in noncommutative rings", second edition, Graduate Texts in Mathematics 131, Springer-Verlag, 2001.
H. Matsumura: "Commutative Ring Theory", Cambridge University Press, 1989.
R.S. Pierce: "Associative algebras", Graduate Texts in Mathematics 88, Springer-Verlag, 1982.
J.P. Serre: "Linear representations of finite groups", Graduate Texts in Mathematics 42, Springer-Verlag, 1977""""""""""""""