# General Algebra

## Objectives

It is intended that students acquire knowledge on

I. Order, lattices, ordered algebraic structures;

II. Introduction to Commutative Algebra;

III. Elements of Category Theory.

## General characterization

##### Code

11584

##### Credits

9.0

##### Responsible teacher

Herberto de Jesus da Silva

##### Hours

Weekly - 4

Total - 60

##### Teaching language

Português

### Prerequisites

Elementary knowledge of Group Theory and Ring Theory customarily provided in a Mathematics degree.

### Bibliography

1. B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, 2nd Edition, Cambridge University Press, 2002.

2. Atiyah and MacDonald, Introduction to Commutative Algebra, Addison Wesley Publishing, 1994.

3. Hoffman, Jia and Wang, Commutative Algebra: An Introduction, Mercury Learning & Information, 2016.

4. S. Mac Lane, Categories for the Working Mathematician (Graduate Texts in Mathematics 5)(second ed.), Springer, Berlin (1998).

5. Tom Leinster, Basic Category Theory, Cambridge Studies in Advanced Mathematics, 2014.

### Teaching method

Classes consist on an oral explanation of the theory which is illustrated by examples and the resolution of some exercises.

### Evaluation method

There are two mid-term tests. These tests can substitute the final exam if CT is, at least, 9.5. CT is the arithmetic mean of the non-rounded grades of the tests.

To be approved in final exam, the student must have a minimum grade of 9.5 in it.

More detailed rules are available in the portuguese version.

## Subject matter

I. Order, lattices, ordered algebraic structures: ordered sets; lattices and complete lattices; modular, distributive and Boolean lattices.

II. Introduction to Commutative Algebra: Prime ideals and maximal ideals; NiIradical and Jacobson radical; Operations on ideals; Rings and modules of fractions; Primary decomposition.

III. Elements of Category Theory: Definition and examples of categories; Functors and natural transformations; Equivalence of categories; Products and coproducts; The Hom functors; Representable functors.