Academic year 2022-23: summary of the lectures and links to the
videos (if not expired)
- Lecture 1. (2022/10/7) Definition of a (commutative,
unitary ring, definition of an ideal of a ring, quotient of a
ring w.r.t. an ideal. Finitely generated ideals,
principal ideals. Ring homomorphism. Theorems of homomorphism.
Zero divisors in a ring, invertible elements in a ring.
Integral domains.
Lecture
- Lecture 2. (2022/10/12) Law of the double quotient for
rings and ideals. Modules over a (commutative, unitary) ring.
Submodules, quotient modules, homomorphisms of modules. Product
of modules, (external) direct sum of modules. Universal property
of the product and of the direct sum. Free modules.
Basis of a module. Examples.
Part 1
Part 2
- Lecture 3. (2022/10/14) Partially order sets: equivalence
of the chain condition, the finite chain condition and the existence
of a maximal element. Definition of artinian and noetherian modules
and rings. A module is noetherian if and only if all its submodules
are finitely generated. Exact sequences and short exact sequences.
In a short exact sequence the central module is artinian/noetherian
iff the two extreme modules are artinian/noetherian.
Part 1
Part 2
- Lecture 4. (2022/10/19) Noetherian and artinian modules:
a finite direct sum of noetherian/artinian modules is a
noetherian/artinian module. If a ring is noetherian/artinian, a
finitely generated module over the ring is noetherian/artinian.
Examples of rings which are not noetherian.
Hilbert Basis Theorem. Sum of modules and internal direct sum of
modules. Definition of PID. Definition of prime and irreducible
element in a domain. Prime implies irreducible.
Part 1
Part 2
- Lecture 5. (2022/10/21) Greatest common divisor in PID.
Bezout identity. In a PID irreducibel = prime. Definition of UFD.
A PID is a UFD.
Cyclic modules over PID. First properties of cyclic modules.
Part 1
Part 2
- Lecture 6. (2022/10/26) Some preparatory lemmas
regarding extension of homomorphisms. Torsion elements in a
module. Definition of a torsion module. Minimal annihilator
of a module. A module over a PID has an element of order equal
to the minimal annihilator of the module. Theorem of cyclic
decompsition of a f.g. torsion module.
Uniqueness of the decomposition.
Part 1
Part 2
- Lecture 7. (2022/10/28) Uniqueness of the cyclic
decomposition of a f.g. torsion module over a PID. First class
of examples: Classification of finite abelian groups. Correspondence
between vector spaces over a field K plus endomorphism and
modules over K[x]. In the correspondence, finite dim
vector spaces correspond to f.g. torsion modules (and
conversely).
Part 1
Part 2
- Lecture 8. (2022/11/2) In the correspondence defined in
lecture 7 cyclic modules correspond to vector spaces generated
by elements of the form t^i(c) (i = 0, 1, ..., n).
Companion matrix of a monic polynomial. If (V, t) is a
vector space + an endomorphism, there exists a basis in V
for which t is represented by a matrix which is the direct
sum of companion matrices. Cayley-Hamilton theorem.
Part 1
Part 2
- Lecture 9. (2022/11/9) Examples of cyclic decomposition
of modules. Primary modules. Theorem of primary decompostion of
modules over a PID. Decomposiiton of a module over a PID in
cyclic primary modules. Elementary divisors of a module.
First application: decomposition of finite abelian groups into cyclic
primary modules. Elementary Jordan matrices.
Part 1
Part 2
- Lecture 10. (2022/11/11) Jordan canonical form of a
matrix.
Rings in general: a commutative unitary ring has a maximal ideal
(Zorn's lemma). Free modules. Two bases of a free module
have the same cardinality. Rank of a free module. All the submodules
of a free, finitely generated module F over a PID are free of
rank smaller or equal to the rank of F.
Part 1
Part 2
- Lecture 11. (2022/11/16) Modules finite generated torsionless
over a PID are free. Torsion part of a module. Decomposition of a
finitely generated module over a PID into cyclic submodules
(of orders s.t. every order is a multiple of the next one) and a
free part.
Maximal ideals in a ring. Every proper ideal is contained in a
maximal ideal. Local rings.
Part 1
Part 2
- Lecture 12. (2022/11/18) A ring R is local if and
only if there exists an ideal M in R such that
all the elements of the complement of M in R are
invertible. Operations with ideals: sum, intersection, product.
Chinese remainder theorem for ideals. If an ideal is contained
in the union of finitely many primes, then it is contained in one of
the primes. If a prime ideal contains an intersection of ideals,
it contains one of the ideals.
Part 1
Part 2
- Lecture 13. (2022/11/23) Operations on ideals: colon
operation and the radical of an ideal. Extensions and contractions
of ideals (w.r.t. a ring homomorphism). Nakajama lemma and some
consequences. First part of the construction of the ring of fractions.
Part 1
Part 2
- Lecture 14. (2022/11/25) Properties of the ring of
fractions. Modules of fraction. The functor S-1
is exact. Local properties. Extension and
contraction of ideals in the ring of fractions.
Part 1
Part 2
- Lecture 15. (2022/11/30) Properties of the
extension and contractions of ideals in the
ring of fractions. The 1-1 correspondence between prime ideals
of a ring R with empty intersecton with a m.c.s. S
and prime ideals in S-1R. Definition of primary
ideals. Properties of primary ideals. Radical of a primary ideal.
Colon operation of primary ideals. Definition of a primary
decomposition of an ideal.
Part 1
Part 2
- Lecture 16. (2022/12/01) Primary decomposition of an
ideal. Irreduntant (or minimal or reduced) primary decomposition.
First uniqueness theorem of the primary decomposition of an ideal.
Primary ideals and multiplicatively closed subsets: there is a
1-1 correspondence between primary ideals
of a ring R with empty intersecton with a m.c.s. S
and primary ideals in S-1R. Isolated and embedded
components of a primary decomposition. Second theorem
of uniqueness of primary decomposition.
Irreducible ideals. In a noetherian ring every ideal is a finite
intersection of irreducible ideals.
Part 1
Part 2
- Lecture 17. (2022/12/06) In a noetherian ring an irreducible
ideal is primary. In a noetherian ring every ideal has a primary
decomposition. In a noetherian ring a suitable power of the radical
of an ideal I is contained in I. In a noetherian ring
the nilradical is nilpotent. In a noetherian ring the associated
prime to an ideal I are the primes in the set I:(x)
where x varies in the ring. An example of a (necessarily
not noetherian) ring where tahere are ideals which do not admit
a primary decomposition.
Part 1
Part 2
- Lecture 18. (2022/12/07) Some examples of the use of
primary decomposition. Artinian rings. Chain of modules, composition
series, length of a module.
Part 1
Part 2
- Lecture 19. (2022/12/13) A module has finite length iff it
satisfies a.c.c. and d.c.c. If in a ring the zero ideal is a product
of maximal ideal, the ring is artinian if and noly if is notetherian.
In an artinian ring there are only a finite number of maximal ideals.
The Krull dimension of a ring.
A ring is artinian if and only if it is noetherian and of Krull
dimension zero.
Part
1
Part
2
- Lecture 20. (2022/12/14) Every artinian ring is a product of
finitely many artinian local rings. Examples. An example of a diophantine
equation. A summary of the basic notions of field theory.
Part 1
Part 2
- Lecture 21. (2022/12/16) Number fields. Abel theorem
(or theorem of the primitive element). A number field is of the form
Q[a], where a is an algebraic number. Symmetric
polynomials. The fundamental theorem on symmetric polynomials. Basis
of a number field over the rationals. The discriminant of a basis. Properties
of the discriminant (is a rational number). Algebraic integers.
Characterization of algebraic integers.
Part 1
Part 2
- Lecture 22. (2022/12/20)
The set of algebraic integers is a ring (extension of the integers and
a subring of the complez numbers). Definition of the ring of algebraic integers
of a number field. Integral bases. Definition and existence. The ring of
algebraic integers, as a group, is a free abelian group of tank equals to
the degree of the number field. The ring of algebraic integers is noetherian.
Definition of a integral element of a ring S over a ring R.
Integal extension. Integrally closed ring.
The ring of algebraic integers is integally closed and of krull dimension 1.
Definition of Dedekind domain.
Part 1
Part 2
- Lecture 23. (2022/12/21) In a integral extension of two domains one
is a field if and only if the other is a fiels. Consequence: the ring of
algebraic integers of a number filed has Krull dimension one. The ring of
algebraic integers is integrally closed. Properties of Dedekind domains.
Several characterization of Dedekind domains. An example.
Part 1
Part 2