ELEMENTS OF COMMUTATIVE ALGEBRA
Academic year and teacher
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- ALBERTO CALABRI
- Credits
- 6
- Didactic period
- Primo Semestre
- SSD
- MAT/02
Training objectives
- Knowledge of the fundamental concepts of commutative algebra, such as:
- commutative rings,
- quotient rings,
- prime ideals and maximal ideals,
- localizations of a ring,
- primary decomposition of an ideal,
- modules on a ring,
- tensor product of two or more modules,
- noetherianity and artinianity of rings and modules.
At the end of the course the student will be able to recognize:
- examples of commutative rings,
- the properties of the elements of a commutative ring,
- the properties of the ideals of a commutative ring,
- examples of modules on a ring,
- the properties of noetherianity and artinianity of modules and rings,
and will be able to construct new rings and modules using localization and the tensor product. Prerequisites
- First year Algebra.
Course programme
- Rings and ring homomorphisms. Ideals. Quotient rings. Zero-divisors. Nilpotent elements. Units. Prime ideals and maximal ideals. Nilradical and Jacobson radical.
Operations on ideals. Extension and contraction of ideals (14 hours).
Modules and module homomorphisms. Submodules and quotient modules. Operations on submodules. Direct sum and product. Finitely generated modules. Exact sequences. Tensor product of modules. Restriction and extension of scalars. Exactness properties of the tensor product. Algebras (12 hours).
Rings and modules of fractions. Local properties. Extended and contracted ideals in rings of fractions (6 hours).
Hilbert basis theorem. Hilbert theorem of zeroes (4 hours).
Primary decomposition. Chain conditions. Noetherian Rings. Artin Rings (6 hours).
Integral dependence. The going-up theorem. Integrally closed integral domains. The going-down theorem.
Discrete valuation rings. Dedekind domains (6 hours). Didactic methods
- Lessons at the blackboard.
Learning assessment procedures
- Oral exam consisting of three questions on general aspects of the course: for at least one question, a proof with some details is required.
The answer to a question is evaluated with a score from 0 to 10, depending on the clarity and precision of the response, and the final score is the sum of the three scores.
In order to pass the exam, the score must be at least 18. Reference texts
- M.F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.
