Short summary of the lectures (a.y. 2021-22) and Video recording

• Lecture 1. (2021/10/4) Definition of rings, ideals, quotient rings, homomorphisms of rings, theorems of homomorphisms...
  Video 1
• Lecture 2. (2021/10/5) Modules over a ring. Product of modules, direct sum of modules.
  Video 2
• Lecture 3. (2021/10/11) Free modules, basis of a free module, finitely generated modules, examples of modules, abelian groups are the same as modules over the ring of integers. Every module is a quotient of a free module, exact sequences of modules, short exact sequences. Partially ordered sets, definition of noetherian and artinian modules. First examples.
  Video 3A, Video 3B.
• Lecture 4. (2021/10/12) Properties of noetherian and artinian modules. Noetherian and artinian rings. Examples of rings (modules) which are not noetherian. Hilbert basis theorem.
  Video 4 (sorry, first part of the lecture... forgotten to record).
• Lecture 5. (2021/10/18) Consequences of Hilbert Basis Theorem. Definition and properties of PID's (Principal Ideal Domains). The gcd in a PID. Bezout identity. In a PID prime = irreducible. Definition of UFD (Unique Factorization Domanis)A PID is a UFD.
  Video 5A, Video 5B.
• Lecture 6. (2021/10/19) Modules over PID. Cyclic modules. Order of a cyclic module. Some lemmas. Annihilator of a module. Minimal annihilator. Torsion elements. Torsion modules. Construction of an element of order equals to the minimal annihilator of a module.
  Video 6A, Video 6B.
• Lecture 7. (2021/10/25) Decomposition of a module over a PID into the direct sum of cyclic modules. Unicity of the decomposition. Examples: Classification of finite abelian groups. A vector space over a field K with an endomorphism give a K[x] module, and conversely.
  Video 7A, Video 7B
• Lecture 8. (2021/10/26) More details on the correspondence between vector spaces plus endomorphism and modules over the ring of polynomials. Invariant vectors subspaces, cyclic vector subspaces. Companion matrices of a polynomial.
  Video 8A, Video 8B.
• Lecture 9. (2021/11/08) The decomposition of a f.g. torsion module into cyclic modules (over a PID) gives a that a square matrix is similar to the direct sum of companion matrices. Cayley-Hamilton theorem.
  Video 9A, Video 9B.
• Lecture 10. (2021/11/09) Primary modules. Decomposition of a module over a PID into primary modules (Primary decomposition of a module). Decomposition of a module over a PID into cyclic, primary modules. Jordan canonical form of a matrix.
  Video 10A, Video 10B.
• Lecture 11. (2021/11/15) Zorn's lemma. Existence of maximal ideals in a unitary ring. Free modules over a ring. Rank of a free module. Short exact sequences and split exact sequences.
  Video 11A, Video 11B.
• Lecture 12. (2021/11/16) Torsion free modules. If R is a PID, then submodules of free f.g. modules are free. Torsion submodule of a module. Decompositon of a f.g. module over a PID as a direct sum of cyclic modules and a free module. Finite presentation of a f.g. module over a noetherian ring.
  Video 12A, Video 12B.
• Lecture 13. (2021/11/22) From a finite presentation of a module, it is possible to identify a finitely generated module over a noetherian ring as a matrix with entries in the ring. A particular case: The Smith normal form.
  Commutative rings: ideals, prime ideals, maximal ideals, local rings. Nilradical of a ring.
  Video 13A, Video 13B (error of Teams...).
• Lecture 14. (2021/11/23) Operation with ideals. Coprime ideals, the Chinese Remainder Theorem (for rings). If an ideal is contained in a union of prime ideals, it is contained in one of the primes.
  Video 14A (error of Teams...), Video 14B, Video 14C.
• Lecture 15. (2021/11/29) If a prime contains an intersection of ideals, it contains one of the ideals of the intersection. Colon operation. Radical of an ideal, extensions and contractions of ideals. Nakajama lemma.
  Video 15A, Video 15B.
• Lecture 16. (2021/11/30) Rings of fractions. Properties of the rings of fractions. Localizations, fractions of modules.
  Video 16A, Video 16B.
• Lecture 17. (2021/12/6) The functor S^-1 is exact. Local properties of modules. Ideals extended and contracted in the case of the ring of fractions.
  Video17A, Video 17B.
• Lecture 18. (2021/12/7) Primary ideals. Examples. Radical of a primary ideal. P-primary ideals. Some preparatory lemmas on primary ideals. Primary decomposition. Definition of a minimal primary decomposition.
  
• Lecture 19. (2021/12/13) Primary ideals and multiplicatively closed sets. Uniqueness of primary decomposition of an ideal: First and second theorem.
  Video 19A, Video 19B.
• Lecture 20. (2021/12/14) Irreducible ideals. In a noetherian ring every irreducible ideal is primary. In a noetherian ring, every ideal is a finite intersection of irreducible ideals. As a consequence, we have that in a noetherian ring, every ideal has a (minimal) primary decomposition. Examples of primady decomposition of ideals (see also the following two files: Example 1 and Example 2
  Video 20A, Video 20B.
• Lecture 21. (2021/12/20) Length of a module. Composition series of modules. Modules of finite length. Artinian rings. In an artinian ring all the prime ideals are maximal. An artinian ring has only a finite number of maximal ideals. In an artinian ring the nilradical is nilpotent.
  Video 21A, Video 21B.
• Lecture 22. (2021/12/21) Definition of Krull dimension of a ring. A ring is artinian if and only if is noetherian and of Krull dimension zero. Local artinian rings. Examples of artinian rings. An artinian ring is isomorphic (in a unique way) to a product of local artinian rings.
  Video 22A, Video 22B.
• Lecture 23. (2022/01/10) Height of an ideal, Krull dimension, Krull's principal ideal (without proof), some results on the height of ideals. Graded rings and modules, homogeneous multiplicatively closed subsets of a ring. Noetherian graded rings.
  Video 23A, Video 23B.
• Lecture 24. (2022/01/11) Associated graded ring A to a ring w.r.t. an ideal of A. The Rees algebra. Primes associated to a quotient M/N of modules. Theorem of Matijtvich-Roberts (without proof). Dimension of a non negative graded ring. Existence of particular chains of graded submodules of a graded module M (if the base ring is noetherian).
  Video 24A, Video 24B.
• Lecture 25. (2022/01/17) Hilbert function and its properties.
  Video 25A Video 25B
• Lecture 26. (2022/01/18) Applications.
  Video 26A Video 26A