# Matematica | Fundamentals of Higher Algebra

## Matematica Fundamentals of Higher Algebra

 0522200009 DIPARTIMENTO DI MATEMATICA EQF7 MATHEMATICS 2022/2023

 YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2018 AUTUMN SEMESTER
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/02 6 48 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 PATRIZIA LONGOBARDI T
ExamDate
ISTITUZIONI DI ALGEBRA SUPERIORE13/06/2023 - 12:00
ISTITUZIONI DI ALGEBRA SUPERIORE11/07/2023 - 12:00
Objectives
THE AIM OF THIS COURSE IS TO CONTINUE THE STUDY, ALREADY STARTED DURING THE COURSES OF ALGEBRA I AND II, 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, ALGEBRA II.
Contents
TOPICS IN GROUP THEORY (24 HOURS):

- THE CONJUGACY IN A GROUP.
- DOUBLE COSETS, FROBENIUS THEOREM.
- CENTRALIZERS, NORMALIZERS, THE CENTRE OF A GROUP.
- RING OF ENDOMORPHISMS AND GROUP OF AUTOMORPHISMS OF A GROUP. CHARACTERISTIC SUBGROUPS, FULLY INVARIANT SUBGROUPS.
- CONSTRUCTION OF A GROUP FROM A COMMUTATIVE REGULAR SEMIGROUP.
- DIHEDRAL GROUPS.
- PERMUTATION GROUPS, CYCLES; CYCLIC STRUCTURE OF A PERMUTATION OVER A FINITE SET, CONJUGATE PERMUTATIONS, THE ALTERNATING GROUP A_N, ITS SEMPLICITY FOR N > 4.
- DIRECT PRODUCTS.
- GROUPS OF ORDER < 9.
- P-GROUPS. PRÜFER GROUPS. SYLOW SUBGROUPS OF A GROUP. SYLOW THEOREMS. PROPERTIES OF FINITE P-GROUPS.
- FINITELY GENERATED ABELIAN GROUPS.
- COMMUTATORS AND DERIVED SUBGROUP OF A GROUP. METABELIAN GROUPS. ITÔ'S THEOREM. SOLUBLE GROUPS.
- THE FRATTINI SUBGROUP.

TOPICS IN RING THEORY (24 HOURS):

- DIRECT PRODUCTS (DIRECT SUMS) OF RINGS.
- THE CHINESE REMAINDER THEOREM FOR RINGS.
- MAXIMAL IDEALS, PRIME IDEALS, PRIMARY IDEALS.
- LOCAL RINGS.
- RINGS OF FRACTIONS. LOCALIZATION OF A RING.
- THE RADICAL OD AN IDEAL. THE NILRADICAL OF A COMMUTATIVE RING. THE JACOBSON RADICAL OF A RING.
- NIL IDEALS, NILPOTENT IDEALS.
- CHAIN CONDITIONS: NOETHERIAN RINGS, ARTINIAN RINGS, HILBERT'S BASIS THEOREM.
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 REPRINT 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).
- T.W. HUNGERFORD, ALGEBRA, SPRINGER-VERLAG, BERLIN, 1974,
- N. JACOBSON, BASIC ALGEBRA I, II, FREEMAN, SAN FRANCISCO, 1980.