# Matematica | COMMUTATIVE ALGEBRA

## Matematica COMMUTATIVE ALGEBRA

 0512300048 DIPARTIMENTO DI MATEMATICA EQF6 MATHEMATICS 2022/2023

 YEAR OF COURSE 3 YEAR OF DIDACTIC SYSTEM 2018 SPRING SEMESTER
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/02 6 48 LESSONS OPTIONAL SUBJECTS
 MARIA TOTA T
Objectives
THE AIM OF THIS COURSE IS TO CONTINUE THE STUDY, ALREADY STARTED DURING THE COURSES OF ALGEBRA I/II AND ALGEBRA III, OF TWO RELEVANT CLASSES OF ALGEBRAIC STRUCTURES: RINGS AND GROUPS. EXAMPLES AND APPLICATIONS WILL HELP STUDENTS TO BE ACQUAINTED TO THESE THEORIES, TO THEIR TECHNIQUES, TO THEIR MOTIVATIONS, ALSO IN VIEW OF POSSIBLE FUTURE DEVELOPMENTS.
Prerequisites
GOOD KNOWLEDGE OF THE SUBJECTS CONTAINED IN THE CLASSES OF ALGEBRA I/II AND ALGEBRA III.
Contents
TOPICS IN GROUP THEORY:

- DIRECT SUMS OF GROUPS.
- FINITE ABELIAN GROUPS.
- FREE ABELIAN GROUPS.
- DIVISIBLE ABELIAN GROUPS.
- ABELIAN GROUPS WITH MAX OR MIN.

TOPICS IN RING THEORY:

- CHAIN CONDITIONS.
- NOETHERIAN RINGS.
- ARTINIAN RINGS.
- DEDEKIND RINGS.
Teaching Methods
LECTURES. ATTENDANCE TO CLASS LESSONS IS STRONGLY RECOMMENDED.
Verification of learning
THE AIM OF THE EXAMINATION IS TO EVALUATE THE FAMILIARITY OF THE STUDENT WITH SOME TOPICS IN GROUP THEORY AND IN RING THEORY.
THE EXAMINATION IS ORAL. THE STUDENT HAS TO TALK ABOUT EXAMPLES, CONSTRUCTIONS AND THE PRINCIPAL PROPERTIES OF SOME CLASSES OF GROUPS AND OF RINGS. HE HAS TO SOLVE SOME EXERCISES.
Texts
- M. CURZIO, P. LONGOBARDI, M. MAJ - LEZIONI DI ALGEBRA , LIGUORI, 1994, I RISTAMPA 1996, II ED. 2014.
- M. F. ATIYAH, I. G. MACDONALD - INTRODUZIONE ALL’ALGEBRA COMMUTATIVA, FELTRINELLI, MILANO, 1981 (INTRODUCTION TO COMMUTATIVE ALGEBRA, ADDISON WESLEY, READING MASS.,1969).
- L. FUCHS – ABELIAN GROUPS, SPRINGER, 2015.
- D. J. S. ROBINSON – A COURSE IN THE THEORY OF GROUPS (II ED.), SPRINGER-VERLAG, NEW-YORK, 1996.
- T. W. HUNGERFORD - ALGEBRA, SPRINGER-VERLAG, BERLIN, 1974.
- N. JACOBSON - BASIC ALGEBRA I, II, FREEMAN, SAN FRANCISCO, 1980.