Commutative Algebra

Code: MAT14347M

6 ECTS

Duration: 15 weeks/156 hours

Scientific Area: Mathematics

Teaching languages: Portuguese

Languages of tutoring support: Portuguese

Regime de Frequência: Presencial

Presentation

In this CU we study commutative rings, their ideals and modules. For a good understanding of these contents it is necessary basic knowledge about Rings and Groups Theory.

Sustainable Development Goals

4. Educação de qualidade - Assegurar a educação inclusiva, e equitativa e de qualidade, e promover oportunidades de aprendizagem ao longo da vida para todos.

Learning Goals

- Knowledge: having contact with the main objects in commutative algebra and experiencing problems in this context, applying general algebraic techniques and creating strategies that are specific to each problem; exploring applications of commutative algebra in other fields of mathmatics.
- Skills and Competences: developing abstract reasoning and the capacity of finding strategies to solve new problems.

Modules over commutative rings.
Morphisms of modules, submódules and quotient module.
Direct sum and direct product.
Finitely generated modules.
Exact sequences of modules.
Tensor product. Local rings and rings od fractions.
Noetharian rings and Artinian rings.

Teaching Methods

Problem-solving sessions, where students are invited to work on their own or in small groups, with some moments of exposition or discussion involving the whole class.
The evaluation may be either continuous, done through two or three oral presentations, done preferably during the classes, weighting 100% of the classification, the number of which is to be defined by the professor who is responsible for the course unit in each academic year, or by a final exam.
Formative evaluation is done in or by tasks to be done outside class, to improve the learning process; the elements of formative evaluation will have no weight on the final mark.