Commutative Algebra

Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings.

For an introductory course on commutative algebra, one may choose the following topics to teach:

  • Rings and ring homomorphisms

  • Ideals and quotient rings

  • Zero-divisors, nilpotents, and units in a ring

  • prime and maximal ideals

  • Operations on ideals

  • Extensions and contractions

  • Modules and module homomorphisms

  • Submodules and quotient modules

  • Operations on submodules

  • Direct sum and product

  • Exact sequences

  • Tensor product of modules

  • Restriction and extension of the scalar

  • Exactness properties of the tensor products

  • Algebras and their tensor products

  • Rings and modules of fractions

  • Primary decomposition

  • Chain conditions, Noetherian and Artinian rings

  • Valuation and discrete valuation rings

  • Integral dependence, integrally closed integral domains, the going-up, and the going-down theorems

  • Fractional and invertible ideals, Dedekind domains

For more details, one may refer to the following book:

  • Atiyah, M. F., MacDonald, I. G. (2018). An Introduction to Commutative Algebra, CRC Press

For applications of commutative algebra, one may refer to the following books:

  • Cozzens, M., Miller, S. J. (2013). The Mathematics of Encryption (Vol. 29). American Mathematical Soc.

  • Blahut, R. E. (2012). Algebraic methods for signal processing and communications coding. Springer Science & Business Media.

For more helpful resources on commutative algebra see Commutative Algebra Books.

See also the following pages: