Commutative algebra

2022/2023
Programme:
Mathematics, Second Cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
core mandatory
Group:
M2
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

Basics:
Commutative ring, spectrum. Nilradical and Jacobson radical.
Modules, submodules and homomorphisms. Module operations, direct sum and product. Finitely generated modules. Exact sequences. Tensor product of modules and its exactness properties. Restriction and extension of scalars. Algebras and their tensor products.
Noetherian rings, Hilbert's Basis theorem. Noetherian normalization theorem.
Hilbert's Nullstellensatz, Zariski topology.
Rings of fractions, localization.
Primary decomposition. Associated prime ideals, primary components, uniqueness theorems.
Optional themes:
Valuation rings.
Filtration. Artin-Rees lemma.
Completion and Hensel's lemma.
Introduction to the dimension theory.
Polynomials, Gröbner bases.

Readings

M. Reid: Undergraduate Commutative Algebra, Cambridge Univ. Press, Cambridge, 1995.
M. F. Atiyah, I. G. MacDonald: Introduction to Commutative Algebra, Addison-Wesley, Reading, 1994.
D. Cox, J. Little, D. O'Shea: Ideals, Varieties and Algorithms : An Introduction to Computational Algebraic Geometry and Commutative Algebra, 2nd edition, Springer, New York, 2005.
N. Lauritzen: Concrete Abstract Algebra: From Numbers to Gröbner Bases, Cambridge University Press, Cambridge, 2003.

Objectives and competences

The student learns the basics of the theory of commutative algebra and upgrades notions and theories that were met during the undergraduate algebraic courses. The knowledge is consolidated by homeworks and individual problem solving exercises.

Intended learning outcomes

Knowledge and understanding: Learning the basic notions and theorem of commutative algebra and recognizing the concepts in other areas of mathematics.
Application: In algebraic geometry and algebraic number theory.
Reflection: Understanding the theory on the basis of examples and applications.
Transferable skills: Formulations of problems in appropriate language, solving and analysing the results on examples, recognizing algebraic structures in geometry and number theory.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

Homeworks
Written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

David Dolžan:
DOLŽAN, David, OBLAK, Polona. The zero-divisor graphs of rings and semirings. International journal of algebra and computation, ISSN 0218-1967, 2012, vol. 22, iss. 4, 1250033 (20 str.). [COBISS-SI-ID 16312921]
DOLŽAN, David, KOKOL-BUKOVŠEK, Damjana, OBLAK, Polona. Diameters of commuting graphs of matrices over semirings. Semigroup forum, ISSN 0037-1912, 2012, vol. 84, no. 2, str. 365-373. [COBISS-SI-ID 16313433]
DOLŽAN, David, OBLAK, Polona. Commuting graphs of matrices over semirings. V: 1st Montreal Workshop on Idempotent and Tropical Mathematics, June 29 to July 3, 2009, University of Montreal, Canada. Special Issue dedicated to 1st Montreal Workshop, (Linear algebra and its applications, ISSN 0024-3795, Vol. 436, iss. 7). Amsterdam [etc.]: Elsevier, 2011, str. 1657-1665. [COBISS-SI-ID 15585113]
Tomaž Košir:
GRUNENFELDER, Luzius, KOŠIR, Tomaž, OMLADIČ, Matjaž, RADJAVI, Heydar. Finite groups with submultiplicative spectra. Journal of Pure and Applied Algebra, ISSN 0022-4049. [Print ed.], 2012, vol. 216, iss. 5, str. 1196-1206. [COBISS-SI-ID 16183385]
BUCKLEY, Anita, KOŠIR, Tomaž. Plane curves as Pfaffians. Annali della Scuola normale superiore di Pisa, Classe di scienze, ISSN 0391-173X, 2011, vol. 10, iss. 2, str. 363-388. [COBISS-SI-ID 15928409]
KOŠIR, Tomaž, OBLAK, Polona. On pairs of commuting nilpotent matrices. Transformation groups, ISSN 1083-4362, 2009, vol. 14, no. 1, str. 175-182. [COBISS-SI-ID 15077977]