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.
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• 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.