University of Trieste - AY 2023/24
Advanced Algebra (Istituzioni di Algebra Superiore)
Course of Study: Mathematics
Alessandro Logar
Here it is possible to find some material concerning the course
in Advanced Algebra. In particular you can find: the program of the
course (of this academic year and of the past years), several links to
books which are good references for the program of the course,
a short summary of each lecture, lectures and video recordings of
the course given in the past years, further material, like some
written lectures and some examples.
All the lectures are recorded and can be found on the ``Teams'' platform.
Academic year 2023-24: summary of the lectures.
- Lecture 1. (2023/09/25) Definition of a (commutative,
unitary ring, definition of an ideal of a ring, quotient of a ring,
theorems of homomorphisms, zero divisors and unitary elements in
a ring. Modules over rings. (Infinite) product of sets.
- Lecture 2. (2023/09/27) Homomorphisms of modules (also
called linear maps). Finitely generated modules.
Product of modules, (external)
direct sum of modules. Universal properties of product and direct
sum of modules. Free modules.
- Lecture 3. (2023/10/02) Every module is a quotient of
a free module. An example of a quotient of ZxZ.
Partially ordered sets. Artinian and noetherian modules. Exact
sequences and short exact sequences.
- Lecture 4. (2023/10/04) More on artinian and noetherian
modules. Internal direct sum of modules. Hilbert basis theorem
and corollaries.
- Lecture 5. (2023/10/09) Rings: Pincipal ideal domanins
(PID) and unique factorization domains (UFD). In a PID we have
the gcd and the Bezout identity. In a PID irreducible is
equivalent to prime. A PID is a UFD. If R is a UFD,
then R[x] is a UFD.
Modules over PID. Definition of cyclic modules.
- Lecture 6. (2023/10/11) Order of a cyclic module. Minimal
annihilator of a module. Torsion modules. Some lemmas which allow
to show that if M is a finitely generated module and
C is a cyclic submodule of M, then, under suitable
hypothesis, M is the direct sum of C and a submodule
L. Theorem of decomposition of a finitely generated, torsion
module into cyclic submodules.
- Lecture 7. (2023/10/16) Proof of the theorem of decomposition
of a module into cyclic modules. Theorem of the unicity of the
decomposition of a module into cyclic modules and proof. Application:
decomposition of finite abelian groups.
- Lecture 8. (2023/10/18) A vector space V
over a field K plus and endomorphism t
of V give a
a K[x]-module, and we have a 1-1 correspondence between
couples (V, t) and K[x]-modules. In this way we can
apply the decomposition of modules to get results regarding
endomorphisms. Theorem of decomposition of the matrix of
an endomorphism into direct sum of companion matrices.
- Lecture 9. (2023/10/23) Characteristic polynomial of a
companion matrix. Relation between the minimal polinomial of an
endomorphism and the minimal annihilator of the module associated
to the endomorphism. The Cayley-Hamilton theorem. Rational canonical
form of a matrix. Definition of primary modules and p-primary
modules. Primary decomposition of a finitely generated, torsion module
over a PID.
- Lecture 10. (2023/10/25) Given a f.g. torsion module over a
PID, it can be decomposed into cyclic modules and each cyclic module
can be decomposed into primary module. In this way it is possible
to obtain the cyclic primary decomposition of a module. Elementary
divisors of a module. As a consequence,every finite abelian
group is the direct sum of finite cyclic groups of order prime
powers. On the other side, every matrix (endomorphism) for which
the characteristic polynomial is a product of linear polynomial
(e.g. if the field is algebraically closed) is similar to a direct
sum of elementary Jordan matrices (theorem of the Jordan canonical
form).
Zorn's lemma. All the bases of two free modules over a unitary,
commutative ring have the same cardinality.
- Lecture 11. (2023/10/30) Split short exact sequences.
Free modules over PID and submodules of free modules over a PID.
Finitely generated torsionless modules over a PID are free.
Decomposition of a f.g. module over a PID. Finite presentation matrix
of a module (over a noetherian ring).
- Lecture 12. (2023/11/6) Smith Normal form of a matrix.
Examples. Operation with ideals. Sum, product, colon operation.
Coprime ideals. Every proper ideal of a commutative, unitary ring
is contained in a maximal ideal. For coprime ideals products and
intersections coincide. Product of rings. Chinese remainder
theorem for rings.
- Lecture 13. (2023/11/8) Proof of the Chinese R.T.
Properties of pime ideals: If an ideal is contained in a finite
union of prime ideals, it is contained in one of the primes; if
a prime ideal contains a finite intersection of ideals, then it
contains one of the ideals. Nilradical (as intersection of prime
ideals or as the set of nilpotent elements). Radical of an ideal
and some of its properties. Extensions and contractions of ideals.
Nakajama Lemma.
- Lecture 14. (2023/11/13) Some applications of the
Nakajama lemma. Construction of the ring of fractions w.r.t. a
multiplicatively closed subset. Universal property of the construction.
Examples of rings of fractions: the localization of a ring w.r.t.
a prime ideal. Construcition of the modules of fraction. The functor
S-1 is an exact functor.
- Lecture 15. (2023/11/15) Proof of the exactness of
S-1. Local properties. Ideals in
S-1(R).
Every ideal is an extension ideal. Characterizaton of
extended-contracted ideals. Prime ideals in S-1(R).
First notions of primary decomposition of ideals. Definition of
primary ideals.
- Lecture 16. (2023/11/20) Properties of primary ideals.
Radical of a primary ideal. Definition of a primary decomposition
of an ideal. Minimal (ore reduced or irridontant) primary
decompositions. First theorem of uniqueness of the primary
decomposition of an ideal. Minimal prime over an ideal.
The minimal prime over an ideal are precisely the prime ideals
of the primary decomposition which are minimal.
- Lecture 17. (2023/11/22) An example of a ring in which
there are ideals that do not admit a primary decomposition.
Primary ideal and multiplicatively closed sets. Second theorem
of uniqueness of primary decomposition. Isolated compoments
and embedded components of a primary decomposition. Examples
of primary decompositions.
- Lecture 18. (2023/11/27) Noetherian rings: in a noetherian
ring every ideal has a primary decomposition: Irreducible ideals
are primary and every ideal is a finite intersection of irreducible
ideals. Chain of modules. Length of a chain of modules. Composition
series of modules. If a module
has a composition series, then all the composition series are of the
same length. Composition series in vector spaces. If in a ring
the ideal (0) is a finite product of maximal ideals, then
the ring is artinian if and only if is noetherian.
- Lecture 19. (2023/11/29) Artinian rings. If a ring is
artinian and a domain, then is a field. A prime ideal in an
Artinian ring is maximal. An Artinian ring is semilocal
(i.e. has only a finite number of maximal elements). The
nilradical of an Artinian ring is nilpotent. Definition of
Krull dimension of an ideal. A ring is artinian if and
only if is noetherian of Krull dimension zero.
- Lecture 20. (2023/12/4) Structure theorem of artinian rings:
every artinian ring is a finite product of artinian local rings. The
number of factors and the factors are unique (up to isomorphisms and
permutations). Examples. Algebraic numbers, numbers fields.
- Lecture 21. (2023/12/6) Abel's theorem of the primitive
element. Algebraic numbers over a field. Every algebraic number field
is a simple extenson of the field of rationals (and conversely).
Homomorphisms from a simple extension of the rationals and
C. Symmetric polynomials, fundamental theorem of symmetric
polynomials.
- Lecture 22. (2023/12/11) A basis of an algebraic extension.
Discriminant of a basis. Change of basis and corresponding
discriminants. The discriminant of a basis is a rational
number. Integral elements (or algebraic integers). The set of
algebraic integers is a ring. Definition of the ring of algebraic
integers. If K is a number field, it is the quotient
field of its ring of algebraic integers.
- Lecture 23. (2023/12/13) Integral bases. Existence of an
integral basis. The ring of algebraic integers as a group w.r.t. the
addition, is a free abelian group of rank equals to the degree of
its algebraic extension. The ring of algebraic integer is noetherian.
Interal extensions of rings. Integral closure of a ring
(in a bigger ring). Integrally closed ring.
The set of integral elements is a ring. The ring
of algebraic integers is integrally closed.
-
- Lecture 24. (2023/12/18) In a Dedekind domain every ideal
is a product of primary ideals and every primary ideal is a product
of prime ideals. An example: The number field
Q[a], where a is the square root of -5
and its ring of algebraic integers. Its construction. The ring is
not a UFD. Definition of a discrete valutation of a field.
Discrete valutation rings (DVR): definition and some properties.
- Lexcture 25. (2023/12/20) Some other properties of discrete
valutations rings. Characterizations of DVR. Discussion of some
properties of Dedekind domains.
Academic year 2020-21: summary of the lectures and video recording:
See here
Academic year 2021-22: summary of the lectures and video recording:
See here
Academic year 2022-23: summary of the lectures and video recording (if not expired):
See here
Useful material
Course summary
Useful links
Some lectures (some in Italian, some in English)
Examples (some in Italian, some in English)