**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: