Unit of study_

# MATH4312: Commutative Algebra

### 2024 unit information

Commutative Algebra provides the foundation to study modern uses of Algebra in a wide array of settings, from within Mathematics and beyond. The techniques of Commutative Algebra underpin some of the most important advances of mathematics in the last century, most notably in Algebraic Geometry and Algebraic Topology. This unit will teach students the core ideas, theorems, and techniques from Commutative Algebra, and provide examples of their basic applications. Topics covered include affine varieties, Noetherian rings, Hilbert basis theorem, localisation, the Nullstellansatz, ring specta, homological algebra, and dimension theory. Applications may include topics in scheme theory, intersection theory, and algebraic number theory. On completion of this unit students will be thoroughly prepared to undertake further study in algebraic geometry, algebraic number theory, and other areas of mathematics. Students will also gain facility with important examples of abstract ideas with far-reaching consequences.

## Unit details and rules

#### Science

 Prerequisites: ? None None None Familiarity with abstract algebra, e.g., MATH2922 or equivalent

At the completion of this unit, you should be able to:

• LO1. Demonstrate an understanding of key concepts in commutative algebra and its connections to algebraic geometry.
• LO2. Apply these concepts to solve qualitative and quantitative problems in mathematical contexts, using appropriate mathematical techniques as necessary.
• LO3. Distinguish and compare the properties of different types of rings, analysing them into constituent parts.
• LO4. Formulate geometric problems and algebraic terms and determine the appropriate framework to solve them.
• LO5. Synthesise knowledge from fundamental theorems in commutative algebra and use this to prove new results.
• LO6. Demonstrate a broad understanding of important concepts in commutative algebra and exercise critical thinking in recognising and using these concepts to draw conclusions and analyse examples.
• LO7. Communicate coherent mathematical arguments appropriately to student and expert audiences, both orally and through written work.

## Unit availability

This section lists the session, attendance modes and locations the unit is available in. There is a unit outline for each of the unit availabilities, which gives you information about the unit including assessment details and a schedule of weekly activities.

The outline is published 2 weeks before the first day of teaching. You can look at previous outlines for a guide to the details of a unit.

Session MoA   Location Outline
Semester 2 2024
Normal day Camperdown/Darlington, Sydney
Outline unavailable
Session MoA   Location Outline
Semester 1 2020
Normal day Camperdown/Darlington, Sydney
Semester 2 2021
Normal day Camperdown/Darlington, Sydney
Semester 2 2021
Normal day Remote
Semester 2 2022
Normal day Camperdown/Darlington, Sydney
Semester 2 2022
Normal day Remote
Semester 2 2023
Normal day Camperdown/Darlington, Sydney

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### Modes of attendance (MoA)

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