Math 676: Commutative Algebra.

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Catalog Information


Commutative Algebra.

Credit Hours



Math 572


Commutative rings, modules, tensor products, localization, primary decomposition, Noetherian and Artinian rings, application to algebraic geometry and algebraic number theory.

Desired Learning Outcomes

Minimal learning outcomes

Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the concepts below, related to, but not identical to, statements proven by the text or instructor.

  • Commutative rings and ideals
  • Modules
  • Tensor products
  • Localization
  • Primary decomposition
  • Integral dependence
  • Noetherian and Artinian rings
  • Dedekind domains and discrete valuation rings
  • Applications to algebraic geometry


Possible textbooks for this course include (but are not limited to):

  • Atiyah & Macdonald, Introduction to Commutative Algebra
  • Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry

Additional topics

Possible additional topics are:

  • Completions
  • Dimension theory

Courses for which this course is prerequisite

Math 663

Math 664