Chiara NICOTERA | Fundamentals of Higher Algebra
Chiara NICOTERA Fundamentals of Higher Algebra
cod. 0522200009
FUNDAMENTALS OF HIGHER ALGEBRA
| 0522200009 | |
| DEPARTMENT OF MATHEMATICS | |
| EQF7 | |
| MATHEMATICS | |
| 2024/2025 |
| YEAR OF DIDACTIC SYSTEM 2018 | |
| AUTUMN SEMESTER |
| SSD | CFU | HOURS | ACTIVITY | |
|---|---|---|---|---|
| MAT/02 | 6 | 48 | LESSONS |
| Exam | Date | Session | |
|---|---|---|---|
| ISTITUZIONI DI ALGEBRA SUPERIORE | 08/01/2025 - 10:00 | SESSIONE ORDINARIA | |
| ISTITUZIONI DI ALGEBRA SUPERIORE | 08/01/2025 - 10:00 | SESSIONE DI RECUPERO | |
| ISTITUZIONI DI ALGEBRA SUPERIORE | 28/01/2025 - 10:00 | SESSIONE ORDINARIA | |
| ISTITUZIONI DI ALGEBRA SUPERIORE | 28/01/2025 - 10:00 | SESSIONE DI RECUPERO | |
| ISTITUZIONI DI ALGEBRA SUPERIORE | 19/02/2025 - 10:00 | SESSIONE ORDINARIA | |
| ISTITUZIONI DI ALGEBRA SUPERIORE | 19/02/2025 - 10:00 | SESSIONE DI RECUPERO |
| Objectives | |
|---|---|
| THE AIM OF THIS COURSE IS TO CONTINUE THE STUDY OF TWO RELEVANT CLASSES OF ALGEBRAIC STRUCTURES: GROUPS AND RINGS. 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 THE STUDENT MUST BE ABLE TO RECOGNIZE AND USE THE ALGEBRAIC STRUCTURES STUDIED. FURTHERMORE, HE/SHE MUST BE ABLE TO APPLY GROUP THEORY AND RING THEORY TOOLS TO OTHER AREAS OF SCIENCES. TEACHING WILL AIM TO PROMOTE STUDENTS' AUTONOMY OF JUDGMENT SO THAT THEY ARE ABLE TO CRITICALLY ANALYZE THE CONCEPTS STUDIED. STUDENT WILL BE STRONGLY ENCOURAGED TO TAKE CARE OF THE PRESENTATION SO THAT THEY ARE ABLE TO PRESENT THE ARGUMENTS CLEARLY AND USING APPROPRIATE LANGUAGE. STUDENTS WILL BE ABLE TO IMPROVE AND TO APPLY THEIR STUDY INDIPENDENTLY. |
| Prerequisites | |
|---|---|
| GOOD KNOWLEDGE OF THE SUBJECTS CONTAINED IN THE CLASSES OF ALGEBRA I/II AND ALGEBRA III. |
| 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. BASIC PROPERTIES OF 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. |
| More Information | |
|---|---|
| TEACHER'S EMAIL ADDRESS: CNICOTERA@UNISA.IT |
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