# ELEMENTS OF ADVANCED ALGEBRA

**Academic Year 2017/2018**- 1° Year

**Teaching Staff**

- Topics in algebra:
**Marco D'ANNA** - Computational Algebra:
**Vincenzo MICALE**

**Credit Value:**12

**Scientific field:**MAT/02 - Algebra

**Taught classes:**70 hours

**Exercise:**24 hours

**Term / Semester:**1° and 2°

## Learning Objectives

**Topics in algebra**The aim of this course is to deepen the study of coomutative ring theory, taking particular attention to polynomial rings and their quotients, with a view towards applications to allgebraic geometry and number theory.

The student will also improve his skill of making abrstact argomentations and will learn that a deep theoretical knowledge allows to develop signifinant applicative tools.

**Computational Algebra**The objective of the second module of the course is to introduce the theory of Groebner bases , in order to begin the computational student to algebra and its applications

## Detailed Course Content

**Topics in algebra***I. Rings and ideals.*Definitions and first properites. Prime and maximal ideals. Local rings. Nilradical and Jacobson radical. Ideals operations.Homomorphisms.*II. Modules.*Definitions and first properties. DIrect product and direct sum; free modules. Finite modules and Nakayama's lemma. Module homomorphisms. Algebras.*III. Frctions rings and modules.*Definition and properties. Localization and local properties. Ideals in fraction rings.*IV. Noetherian rings.*Affine varieties, affine K-algebras, correspondance betwwen algebra and algebraic-geometry concepts. Krull dimension. Noetherian rings and modules: definitions and first properties. Hilbert's basis theorem. Conditions for a sub-algebra to be finitely generated.*V. Artinian rings.*Artinian rings and modules. Composition series. Length. A ring is artinian if and only if it is noetherian and zero-dimensional.*VI. Primary decomposition.*Primary ideals and primary decomposition. Associated primes and their characterization. Zero-divisors. Unicity of isolated components. The noetherian case.*VII. Hilbert Nullstellensatz*: weak and strong formulations.*VIII. Integral dependance.*Definitions and first properties. Going Up theorem. Normal domains and Going Down theorem. Noether's normalization lemma.*IX. First steps in dimension theory*. Chain of primes, height, dimension. Krull's principal ideal theorem.Krull's height theorem.Dimension for polynomial rings with coefficient in a field. Local rings. System of parameters. Embedding dimension. Regular local rings (only definition and geometric relevance).**Computational Algebra**I. Basic Theory of Groebner Bases. The linear case. The case of a single variable. Monomial orders. The division algorithm. Definition of Groebner Bases. S - polynomials and Buchberger algorithm. Reduced Groebner bases .

II . Applications of Groebner Bases. Elementary applications of Groebner Bases. Theory of elimination. Polynomial maps. Some applications to Algebraic Geometry .

III . Modules. Groebner bases and Syzygies. Calculation of the module of syzygy of an ideal.

## Textbook Information

**Topics in algebra**1. M.F. Atiyah, I.G. Macdonald , Introduzione to commutative algebra.

2. E. Kunz, Introduction to commutative algebra and algebraic geometry, Birkauser 1985

**Computational Algebra**W.W. Adams, P. Loustaunau, An introduction to Groebner Bases, American Math. Soc, 1994.