Short summary of the lectures (a.y. 2020-21) and Video recording

• Lecture 1. Definition of rings, ideals, quotient rings, homomorphisms of rings, theorems of homomorphism (for rings).
  Video 1
• Lecture 2. Modules over a (commutative, unitary) ring. Submodules, quotient modules, module homomorphism, theorems of homomorphism for modules. Modules generated by a set. Finitely generated modules. Product of modules, external direct sum of modules. Universal property of the direct sum of mosules.
  Video 2
• Lecture 3. Free modules. Universal property of free modules. Basis of a free module. Every module is a quotient of a free module. Exact sequences of modules, short exact sequences. Partially ordered sets. In partially ordered set the following three conditions are equivalent: (1) every strictly increasing chain of elements is finite; (2) every increasing chain of elements is stationary; (3) every non-empty subset of the partially ordered set has a maximal element.
  Video 3A; Video 3B.
• Lecture 4. Artinian and noetherian modules. A module is noetherian if and only if all of its submodules are finitely generated. Given a short exact sequence of modules, the central module is artinian (noetherian) if and only if the two extreme modules of the sequence are artinian (noetherian). A finite direct sum of artinian (noetherian) modules is artinian (noetherian). A quotient of an artinian (noetherian) module is artinian (noetherian). Artinian and noetherian rings. If a ring is artinian (noetherian) then a fintely generated module over the ring is artinian (noetherian). An example of a ring which is not noetherian.
  Video 4A; Video 4B.
• Lecture 5. Hilbert basis theorem and consequences. Internal direct sum of modules. Different elements in a ring: zero divisors, unitary elements, prime elements, irreducible elements, associated elements. Integral domains. In an integral domain prime implies irreducible. Greatest common divisor of two elements of a domain. Example of a domain in which there does not exist the gcd. Principal ideal domains (PID). In a PID the gcd always exists. In a PID irreducible implies prime. Definition of UFD (unique factorization domains). A domain in which every element is a product of irreducible elements is a UFD if and only if irreducible is equivalent to prime.
  Video 5A; Video 5B.
• Lecture 6.Every PID is a UFD. Cyclic modules over a ring that is a PID. Order of a cyclic module. Annihilator of a module. Some technical lemmas which culminates in the following result: if we have a finitely generated module M over a PID and a cyclic submodule C of it, then M is the direct sum of C and another submodule L of M (and L is finitely generated).
  Video 6A; Video 6B.
• Lecture 7. Torsion elements in a module and torsion modules (over a PID). Order of a torsion element. Definition of a minimal annihilator of a finitely generated torsion module. Theorem of decomposition of a finitely generated torsion module into cyclic modules. Invariant factors of a finitely generated torsion modules. Theorem regarding unicity of the decomposition.
  Video 7A; Video 7B.
• Lecture 8. Unicity of the decompositon of a finitely generated, torsion module over a PID into cyclic submodules. An example. Correspondence between couples (V, t) (where V is a K-vector space and t is an endomorphism fo V) and modules over K[x]. In the correspondence, finite dimensional vector spaces go to finitely generated, torsion modules (and conversely). Minimal polynomial of an endomorphism.
  Video 8A; Video 8B.
• Lecture 9. In the correspondence between couples (V, t) (where t is an endomorphism) and K[x] modules, cyclic sub-vector spaces of V go to cyclic sub-modules of M_V, the minimal polynomial of t goes to the minimal annihilator of M_V. Companion matrix of a monic polynomial g. If C is cyclic K[x] module of order g, then it is possible to find a basis in V_C such that the endomorphism t_x is represented by the companion matrix of g. The decomposition theorem of modules into direct sum of cyclic modules gives that if t is an endomorphism of a vector space V, then it is possible to find a basis in V such that the associated matrix to t is the direct sum of companion matrices (uniquely determined) associated to monic polynomials g₁, ... , g_k (invariant factors). Cayley-Hamilton theorem. Two square matrices of the same order are similar if and only if they have the same invariant factors.
  Video 9A; Video 9B.
• Lecture 10. Primary modules (over a PID). Decomposition of a finitely generated, torsion module into primary modules. The cyclic primary decomposition theorem. Elementary divisors. Application to finite abelian group and to endomorphism. A matrix is similar to a matrix which is the direct sum of companion matrices of polynomials of the form p^a, p^b, ... q^m, qⁿ, ... s^u, s^v, ... where p, q, s, ... are monic irreducible polynomials, a is smaller or equal to b , ..., m is smaller or equal to n ..., Elementary Jordan form of a matrix. Here you can find a link to a written version of the lecture (in Italian).
  Video 10A; Video 10B.
• Lecture 11. Jordan canonical form of a matrix. Zorn's lemma. Every (commutative, unitary) ring has a maximal ideal. Free modules over a ring. Basis of a free module. All the bases of a free module have the same cardinality (which is called the rank of the free module). Two free modules are isomorphic if and only if they have the same rank. If we have an epimorphism from a module M to a free module F, then M is decomposed into the direct sum of the kernel of the epimorphism and a free submodule of M isomorphic to F. A submodule of a finitely generated, free module F is free of rank smaller or equal to the rank of M.
  Video 11A; Video 11B.
• Lecture 12. Examples. Here you can find a discussion of some examples, which are also written in the following files:
  • An example of a module over a PID (in English);
  • Un primo esempio con i gruppi abeliani (in Italian)
  • Decomposizione di un modulo con gli invarianti fondamentali (in Italian)
  • Decomposizione di una matrice (in Italian)
  • Esempio riassuntivo (in Italian), (invarianti fondamentali, divisori elementari, forma canonica razionale di una matrice, forma canonica di Jordan).
  Video 12A; Video 12B; Video 12C.
• Lecture 13. Torsionless finitely generated modules over PID are free. Split short exact sequences and their characterization. Decomposition of a finitely generated module over a PID: direct sum of cyclic modules (of orders m₁, ..., m_k such that m_i+1 divides m_i) and a free module of finite rank. Examples.
  Video 13A; Video 13B.
• Lecture 14. Finite presentation of a finitely generated module over a noetherian ring. A fintely generated module is the cokernel of a homomorphism from a free module or rank n to a free module of rank m. Therefore to give a finitely generated module over a noetherian ring is equivalent to give a m x n matrix A with entries in R. To change the basis in the free modules with elementary operators means to transform the matrix A with elementary rows and columns operations. Application to the case R = Z. In this case the matrix A can be transformed into the Smith normal form. From this form, the theorem of decomposition of finitely generated modules into the sum of cyclic modules and a free module is immediate.
  Properties of commutative rings. Local rings.
  Video 14A; Video 14B.
• Lecture 15. The nilradical of a ring. The nilradical as the intersection of all the prime ideals of the ring. The Jacobson radical of a ring (i.e. the intersection of all the maximal ideals of the ring). Operation with ideals: sum, product, intersection. Coprime ideals. The chinese remainder theorem. Properties of prime ideals: if an ideal is contained in a union of prime ideals, then it is contained in one of the prime ideals. If a prime ideal contains an intersection of ideals, then it contains one of the ideals of the intersection. The radical of an ideal. Radical ideals.
  Video 15A; Video 15B.
• Lecture 16. Properties of the radical of an ideal. Colon operation (for ideals). Nakayama lemma and some consequences. Multiplicatively closed subset of a ring a construction of the ring of fraction. Universal property of the ring of fraction.
  Video 16A; Video 16B.
• Lecture 17. Other properties of the ring of fractions. Examples. Localisation of a ring with a prime ideal. A localisation of a ring is a local ring. Fraction of modules. The functor S^-1 is exact. Local properties.
  Video 17A; Video 17B.
• Lecture 18. Connections with ideals in a ring R and in the ring S^-1(R). Primary ideals. Radical of a primary ideal, P-primary ideals. Ideals whose radical is maximal are primary ideals.
  Video 18A; Video 18B.
• Lecture 19. Properties of primary ideals. Definition of primary decomposition of an ideal. Minimal primary decomposition. First theorem of uniqueness: the associated primes to a minimal primary decomposition of an ideal I are uniquely determined by I (are the prime ideals in the set of Rad(I:x) where x varies in the ring. Isolated (minimal) primes and embedded primes of a primary decomposition of an ideal. The isolated primes of an ideal are precisely the minimal primes which contain the ideal.
  Video 19A; Video 19B.
• Lecture 20. Primary ideals and multiplicatively closed subsets. Second theorem of uniqueness of primary decomposition: the primary ideals associated to the isolated primes of a minimal primary decomposition of an ideal are uniquely determined by the ideal. Irreducible ideals. In a noetherian ring, every ideal is a finite intersection of irreducible ideals. In a noetherian ring, every irreducible ideal is primary. As a consequence, in a noetherian ring every ideal admits a (minimal) primary decomposition.
  Video 20A; Video 20B.
• Lecture 21. In a noetherian ring, the nilradical is nilpotent. Chain of modules and composition series of a module. Length of a module. A module has finite length if and only if satisfies acc and dcc. In a vector space, length means dimension. Artinian rings. An artinian domain is a field. In an artinian domain, every prime ideal is maximal.
  Video 21A; Video 21B.
• Lecture 22. Artinian rings. In an Artinian ring the nilradical is nilpontent. Chain of prime ideals; definition of the Krull dimension of a ring. A ring has Krull dimension 0 if and only if prime ideal implies maximal ideal. A ring is artinian if and only if it is noetherian and of Krull dimension zero. Local artinian ring. Every artinian ring is a product (in a unique way) of local artinian rings (proof omitted). Examples.
  Video 22A; Video 22B; Video 22C.
• Lecture 23. Algebraic elements, algebraic extensions, finite extensions of fields, Tower Theorem, Abel's Theorem (or the Theorem of the primitive element). The ring (and field) of algebraic elements (over Q). Definition of number fields. Definition of algebraic integer. Characterization of algebraic integers (in terms of the abelian group generated by the powers of the element). The set of algebraic integer is a ring. The ring O_K of integers of a number field K.
  Video 23A; Video 23B; Video 23C.
• Lecture 24. Examples of ring of integer of a number field. The ring of integer in general is not a UFD. Integral extensions. Integrally closed rings. Fractional ideals. Invertible ideals. Dedekind domains. Several characterizations of Dedekind domains (proofs omitted). Examples of Dedekind domains. In particular, the ring of integers of a number field is a Dedekind domain. A final example.
  Video 24A; Video 24B.