MA316-6-AU-CO:
Commutative Algebra
2023/24
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Autumn
Undergraduate: Level 6
Current
Thursday 05 October 2023
Friday 15 December 2023
15
05 January 2024
Requisites for this module
MA201 and MA204
(none)
(none)
(none)
(none)
Commutative algebra is the cornerstone established by Hilbert to give a formal backing to intuitive arguments in geometry.
This module will provide a solid foundation of commutative rings and module theory, as well as help developing foundational notions helpful in other areas such as number theory, algebraic geometry, and homological algebra. Examples will be key, many of them will be made 'graphic' thanks to Hilbert's Nullstellensatz.
The aims of this module are:
- To introduce students to basic definitions of commutative algebra.
- To develop students’ critical understanding of some main results in commutative algebra.
- To develop understanding of how to apply such results in particular problems.
By the end of the module, students will be expected to:
- Have a systemic understanding of key definitions in the theory of commutative algebra and critical awareness of how they interact and support each other.
- Select and apply relevant theorems to examples.
- Construct arguments to prove properties of rings.
- Solve problems involving homomorphisms between pairs of groups.
- Formulate counterexamples to statements.
- Understand the notion of affine algebraic variety and its relation to the coordinate ring.
- Recognise and work with quotient rings.
- Being able to decide if an exact sequence of modules is exact.
- Apply geometric techniques to obtain and illustrate algebraic properties of particular rings.
Indicative syllabus
Recollection of abstract algebra: rings, fields, integral domains, rings of polynomials and homomorphisms.
Unique Factorisation Domains (UFDs): irreducible elements and factorisation, UFDs, polynomial ring of a UFD is a UFD.
Ideals and arithmetic of ideals: definition of ideal, intersection and sum, finitely generated ideals, principal ideals, principal ideal domains (PIDs), PIDs are UFDs.
Ideals for commutative rings: prime ideals, maximal ideals and quotient of ring by them, existence of maximal ideals using Zorn's lemma.
Prime spectrum of a ring (Spec): relation to geometry. Examples.
Local rings and localisation. Nilradical.
Modules and exact sequences.
Noetherian rings and Noetherian modules. Hilbert's basis Theorem.
Hilbert's Nullstellensatz and affine varieties.
Teaching in the School will be delivered using a range of face to face lectures, classes and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.
The above list is indicative of the essential reading for the course.
The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students.
Further reading can be obtained from this module's
reading list.
Assessment items, weightings and deadlines
| Coursework / exam |
Description |
Deadline |
Coursework weighting |
| Coursework |
Test |
|
|
| Exam |
Main exam: In-Person, Closed Book, 120 minutes during Summer (Main Period)
|
| Exam |
Reassessment Main exam: In-Person, Closed Book, 120 minutes during September (Reassessment Period)
|
Exam format definitions
- Remote, open book: Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.
- In-person, open book: Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.
- In-person, open book (restricted): The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.
- In-person, closed book: The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary,
for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.
Your department will provide further guidance before your exams.
Overall assessment
Reassessment
Module supervisor and teaching staff
Dr Jesus Martinez-Garcia, email: jesus.martinez-garcia@essex.ac.uk.
Dr Jesus Martinez-Garcia
jesus.martinez-garcia@essex.ac.uk
Yes
No
Yes
Prof Stephen Langdon
Brunel University London
Professor
Dr Rachel Quinlan
National University of Ireland, Galway
Senior Lecturer in Mathematics
Available via Moodle
Of 35 hours, 33 (94.3%) hours available to students:
0 hours not recorded due to service coverage or fault;
2 hours not recorded due to opt-out by lecturer(s), module, or event type.
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