Academic year 2024-25: summary of the lectures.

• Lecture 1. (2024/09/24). Introduction to the course. Summary of some basic notions. Rings, ideals, prime and maximal ideals, quotient of rings, ring homomorphisms. Integral domains. Finitely generated ideals.
• Lecture 2. (2024/09/25). Modules over a ring. Basic properties of modules. Submodules, homomorphisms of modules. Product and external direct sum of modules. Fintely generated modules. Free modules.
• Lecture 3. (2024/09/30). More on free modules. Basis for free modules. Internal direct sum of modules. Quotient modules. Examples.
• Lecture 4. (2024/10/01). Exact sequences. Short exact sequences. Artinian and noetherian modules. A module is noetherian iff every submodule of it is finitely generated. Short exact sequences and artinian and noetherian modules. Direct sum of noetherian (artinian) modules is noetherian (artinian).
• Lecture 5. (2024/10/8). Example of a non-noetherian ring/module. Hilbert basis theorem and consequences. Greatest common divisor in a domain. Example of a domain where we do not have gcd. First notions on PID. In a PID we have the gcd. Unique factorization domains.
• Lecture 6. (2024/10/9). A characterization of UFD. A PID is a UFD. The polynomial ring over a UFD is a UFD (without proof). Modules over PID. Cyclic modules. Order of a cyclic module. Torsion elements in a module. Minimal annihilator of a module. Two lemmas on modules over a PID.
• Lecture 7. (2024/10/15). If a module has a cyclic submodule then, with suitable hypotheses, the module is a direct sum of the cyclic module and another submodule of it. Notion of torsion elements, torsion module, minimal annihilator of a module. A torsion module of order m contains an element of order m.
• Lecture 8. (2024/10/16). A finitely generated torsion module is the direct sum of a finite number of cyclic modules of orders such that each order divides the order of the previous one. The decomposition is unique. An application: classification of finite abelian groups.
• Lecture 9. (2024/10/22). About the 1-1 correspondence between modules over a polynomial ring (with coefficient in a field) and vector spaces plus an endomorphism. Companion matrix of a monic polynomial. Invariant factors.
• Lecture 10. (2024/10/23). Proof of the theorem regarding the decomposition of the associated matrix of an endomorphism. Some consequences. The Cayley Hamilton theorem. Invariant factors of a square matrix. Two square matrices of the same order are similar if and only if they have the same invariant factors. Examples.
• Lecture 11. (2024/10/29). Primary modules and p primary modules. Decomposition of a f.g. torsion module into direct sum of primary modules. Decomposition of a f.g. torsion module into cyclic modules of prime order. Elementary divisors of a module. The case of a vector space and an endomorphism. First steps to get the canonical Jordan form of a square matrix.
• Lecture 12. (2024/10/30). The canonical Jordan form of a matrix.
  An application of the Zorn's lemma in order to get that all commutative, unitary rings have a maximal ideal. Free modules over a ring. A basis of a free module. Different basis of a free module have the same cardinality. The rank of a free module.
• Lecture 13. (2024/11/5). Two free modules are are isomorphic if and only if they have bases of the same cardinality. From an epimorphism from a module M onto a free module F, we get a decomposition of M as the direct sum of the kernel of the epimorphism and a free module (isomorphic to F). A submodule of a free module F of finite rank over a PID is free of rank smaller or equal to the rank of F. Finitely generated, torsionless modules over a PID are free. The sub-module of torsion elements of a module.
• Lecture 14. (2024/11/6). Every finitely generated module over a PID, can be decomposed essentially in a unique way into a direct sum of cyclic modules (of orders in a chain of divisions) and a free module of finite rank.
  Ideals in a ring. Operations with ideals. Every proper ideal is contained into a maximal ideal. Coprime ideals. The nilradical (i.e. the set of all the nilpotent elements of a ring). The nilradical is the intersection of all the prime ideals of the ring. Product of rings.
• Lecture 15. (2024/11/12). The Chinese Remainder Theorem (for rings). Properties of prime ideals in a ring. The colon ideal. The radical of an ideal and some of its properties. The radical of an ideal is the intersection of all the primes which contain the ideal. Set of zero divisors of a ring. Examples.
• Lecture 16. (2024/11/13). The Nakajama lemma and some consequences. Multiplicatively closed subsets of a ring. Construction of the ring of fractions.
• Lecture 17. (2024/11/19). Further properties of multiplicative sets. Localisations. Local properties of modules.
• Lecture 18. (2024/11/20). Ideals in S^-1R. Extensions and contractions of ideals in S^-1R. Prime ideals in localisations of rings.
  Primary ideals. Some properties. Intersections of P-primary ideals is P-primary.
• Lecture 19. (2024/11/26). Colon operation on primary ideals. Definition of a primary decomposition of an ideal. Definition of a minimal primary decomposition. First unicity theorem of a minimal primary decomposition of an ideal. Primary decomposition and multiplicatively closed sets.
• Lecture 20. (2024/11/27). Minimal (or isolated) primes and embedded primes of a primary decomposition of an ideal. The minimal primes of a primary decomposition of an ideal I are precisely the minimal primes w.r.t. the inclusion that contain I. In particular, if an ideal has a primary decomposition, then there are only finitely many minimal primes that contain it. Example of a ring where there are ideals wich do not admit a primary decomposition. Second theorem of unicity of primary decomposition. In the noetherian rings all ideals have a primary decomposition.
• Lecture 21. (2024/12/3). Some examples of primary decomposiitons of ideals.
  Composition series of a module. The length of a composition series of a module is an invariant of the module.
• Lecture 22. (2024/12/4). Modules of finite length. A module has finite length iff it is artinian and noetherian. In a vector space the length is the dimension and finite length, finite dimension, acc, dcc are equivalent. If in a ring the zero ideal is a product of maximal ideals, the ring is artinian iff is noetherian. In an artinian ring, prime ideals are maximal. An artinian ring has finitely many maximal ideals. In an artinian ring the nilradical is nilpotent.
• Lecture 23. (2024/12/10). The Krull dimension of a ring. A ring is artinian if and only if is noetherian of Krull dimension zero. An artinian ring is a finite product (in a unique way) of artinian local rings.
  A short introduction to number fields and the ring of integers of a number field.
• Lecture 24. (2024/12/11). Some examples of ring of integers which are not UFD. Dedekind domains (a domain which is noetherian, integrally closed of Krull dimension one). The ring of integers of a number field is a Dedekind domain. An example: The ring Z[a], where a²+5 = 0 .