I.The Basic Information of the Course(Time New Rome, 12 points, bold font)
Course Number: 202012420008
The EnglishName of theCourse：Abstract Algebra (Advanced)
The ChineseName of theCourse：抽象代数II
In-classHours andAllocationtotal class hours: 48, classroom teaching: 40 classhours, classroom discussion: 8 classhours
Credit(s):3
Semester:1
AppliedDiscipline(Professional Degree Category)：Mathematics
CourseObjectOriented:Academic Doctor
EvaluationMode:Closed-book Examination
TeachingMethod:Blended Teaching
CourseOpening Department:Department of Mathematics
Notes:Filling explanation to applied discipline: the general courses like foreign languages, ideological and political theory course may fill in “AllDisciplines”,Mathematics general course may fill inScience andEngineeringDiscipline, and the other courses fill in “Name ofApplicableDiscipline” according to actual teaching objects, and the number of applicable discipline may be more than one.
II.Prerequisite Course( Time New Rome 12 points bold font)
(Time New Rome 10.5 points, line space fixed value 25 pounds, the front and the end of the paragraph 0 line, the same below)
Advanced Algebra, Abstract Algebra (Elementary), General Topology
III. The Objectives and Requirements of the Course(Time New Rome 12 points, total words: about 200 words)
This is a comprehensive and systematic introduction to the basic theories and methods in commutative algebra. This course will build up a foundation for students to further study of algebraic geometry and other frontier areas of pure mathematics. This course focuses on the cultivation of the students’ ability of abstract thinking and logical reasoning. Through studying this course, the students are required to be familiar with the basic theories and methods in commutative algebra, and to have a preliminary understanding of the basic ideas in algebraic geometry. They should be prepared for research in commutative algebra and related areas.
IV. The Content of the Course(Time New Rome, 12 points, bold font, 1000-2000 words)
Description:Other course should reflect the content and cases of “Courses for Ideological and Political education”, besides the Course for Ideological and Political Education.
Section 1 Rings and ideals (6 class hours)
Understanding rings, ring homomorphisms, ideals, quotient rings, zero factors, nilpotent elements, invertible elements, prime ideals, maximal ideals, radicals, operations on ideals, extensions, restrictions, and their properties
Section 2 Modules (6 class hours)
Understanding modules, module homomorphisms, submodules, quotient modules, operations on submodules, direct sums, direct products, finitely generated modules, exact sequences, tensor product of modules, exactness of tensor product, algebra, tensor product of algebras and their properties
Section 3 Division rings and division modules (4 class hours)
Understanding local properties, extensions and restrictions of ideals in fraction rings
Section 4 Primary decomposition (4 class hours)
Understanding primary decomposition
Section 5 Valuation (6 class hours)
Understanding ascending theorem, integrally closed domain, descending theorem, valuation rings and their properties
Section 6 Chain conditions, Notherian rings and Artinian rings (6 class hours)
Understanding ascending chain condition, descending chain condition, Notherian rings, Artinian rings and their properties
Section 7 Discrete valuation ring and Dedekind domain (4 class hours)
Understanding discrete valuation ring, Dedekind domain, fractional ideals and their properties
Section 8 Completion (4 class hours)
Understanding completion, filtration chain, graded ring, graded module and their properties
Section 9 Dimension (4 class hours)
Understanding Hilbert function, the theory of dimensions of Notherian local ring, regular local ring and transcendental dimensions
Section 10 Algebraic varieties and sheaves (4 class hours)
Understanding affine varieties and their basic theories; understanding the theory of sheaves and properties of sheaves.
Courses for Ideological and Political education: Inspire the students to take the endeavor into the research of frontier problems; Inspire them to devote their career into the improvement of Chinese community of algebra and pure mathematics.
V. Reference Books, Reference Literatures, and Reference Materials
A. Text Books, Monographs and References
1. Atiyah, Michael. Introduction to commutative algebra. CRC Press, 2018.
2.Hartshorne, Robin. Algebraic geometry. Vol. 52. Springer Science & Business Media, 2013.
3.杨子胥,高等学校教材：近世代数,高等教育出版社,2011.
B. Learning Resources (Time New Rome 12 points)
1.MIT open courseware:
https://ocw.mit.edu/courses/mathematics/18-703-modern-algebra-spring-2013/
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