# Math 162009 -- Commutative Algebra

**General Information:**

Commutative algebra was born in the 19th century from algebraic geometry, invariant theory, and number theory. Today it is a mature field with activity on many fronts.

Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry

In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings. Considerations related to modular arithmetic have led to the notion of valuation ring. The restriction of algebraic field extensions to subrings has lead to the notions of integral extensions and integrally closed domains as well as the notion of ramification of an extension of valuation rings.

The notion of localization of a ring (in particular the localization with respect to a prime ideal, the localization consisting in inverting a single element and the total quotient ring) is one of the main differences between commutative algebra and the theory of non-commutative rings. It leads to an important class of commutative rings, the local rings that have only one maximal ideal. The set of the prime ideals of a commutative ring is naturally equipped with a topology, the Zariski topology. All these notions are widely used in algebraic geometry and are the basic technical tools for the definition of scheme theory, a generalization of algebraic geometry introduced by Grothendieck.

Many other notions of commutative algebra are counterparts of geometrical notions occurring in algebraic geometry. This is the case of Krull dimension, primary decomposition, regular rings, Cohen–Macaulay rings, Gorenstein rings and many other notions.

**Reading List:**

1-Michael ATIYAH and Ian MACDONALD, "**Introduction to commutative algebra**," Addison-Wesley, 1969.

2-Miles REID, "**Undergraduate commutative algebra**," Cambridge University Press, LMS Student Texts 29, 1995.

3-Sharp, **Steps in commutative algebra**, 2nd edition.

**Class Hours:**

Saturday, (10-12) Room 248

Monday, (16-18) Room 249

**Course Content:** Here is a glimpse of the syllabus you will learn.

**1-Primary Decomposition**

**2-Integral Extensions**

**3-Dimension Theory**

**4-Regular Sequences**

**Grading**

The grading breakdown for this reading course is divided as follows:

20% Oral Presentation

30% Midterm exam

20% Weekly assignments

30% Final exam

**Homework**

Each week, I will assign a set of homework exercises, which will be due the next

meeting. Most of them will be assigned from the textbook.

**Course Notes and other materials:**

After each lecture, I will put the handouts and other useful materials related to current lecture.

Prof. Khashyarmanesh's Lecture Notes on Commutative Algebra

Prof. Naghipoor's Lecture Notes on Commutative Algebra

Prof Naghipoor's Lecture Notes on Homological Algebra