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.