Introduction. Commutative algebra is a branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings are the ring of integer numbers, fields including rational, real, and complex numbers, polynomial rings over arbitrary commutative rings, and rings of continuous functions. The main purpose of this note is introduce and review some helpful books on commutative algebra.
Books on commutative algebra that are helpful for undergraduate students include:
Sharp, R.Y.: Steps in commutative algebra. Vol. 51. Cambridge University Press, 2000.
Watkins, J.J.: Topics in Commutative Ring Theory. Princeton University Press, 2009.
Reid, M.: Undergraduate Commutative Algebra. Vol. 29. Cambridge University Press, 1995.
The above books are intended as an introduction to commutative algebra for students who have taken a basic algebra course while they are not expected to know about ideals, modules, categories, or homological algebra.
For more advanced and brief texts on commutative algebra, one should study the following books:
Atiyah, M.F., Macdonald, I. G.: Commutative Algebra. Addison-Wesley, Reading, Mass (1969).
Kaplansky, I.: Commutative rings. Boston, 1970.
I must add that both the book by Atiyah and Macdonald and the book by Kaplansky appear easy at first glance. However, they can be difficult if the student is not skillful enough at commutative algebra.
Altman and Kleiman's book on commutative algebra is based on the book written by Atiyah & Macdonald. I highly recommend it if you want to have a more modern and recent style of teaching in mathematics.
Altman, A., Kleiman, S.: A term of commutative algebra. Worldwide Center of Mathematics. 441 p. (2013).
In addition, I find the following book to be very helpful. This book is especially very interesting since it has a bit of an algebraic geometry flavor to it, which makes it quite enjoyable to read as well.
Kemper, G.: A Course in Commutative Algebra. Springer Science & Business Media; 2010.
For advanced but more detailed texts on commutative algebra, one should check the following voluminous though classic resources:
Zariski, O., Samuel, P.: Commutative algebra. Springer Science & Business Media, 1975-6.
The following books are very advanced and beneficial reference books on commutative algebra:
Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Vol. 150. Springer Science & Business Media, 1995.
Matsumura, H.: Commutative Ring Theory. Vol. 8. Cambridge University Press, 1989.
The following book covers interesting topics in commutative algebra with their applications in algebraic combinatorics and algebraic topology:
Bruns, W., Herzog, H.J.: Cohen-Macaulay Rings. Cambridge University Press, 1998.
A traditional but very significant approach to commutative ring theory is the multiplicative ideal theory approach. For this approach, the following books are highly recommended:
Gilmer, R. W.: Multiplicative Ideal Theory. Vol. 12. M. Dekker, 1972.
Larsen, M.D., McCarthy, P.J.: Multiplicative Theory of Ideals. Vol. 43. Academic Press, 1971.
Interdisciplinary books on commutative algebra may include:
Gillman, L., Jerison, M.: Rings of Continuous Functions. Princeton, NJ, 1960.
Kreuzer, M., Robbiano, L.: Computational Commutative Algebra. Vol. 1 and 2. Springer Science & Business Media, 2000 & 2005.
Miller, E., Sturmfels, B: Combinatorial Commutative Algebra. Vol. 227. Springer Science & Business Media, 2005.
Stanley, R.P.: Combinatorics and Commutative Algebra. Vol. 41. Springer Science & Business Media, 2007.
In my opinion, a commutative algebraist should be familiar with the following books as well:
Hartshorne, R.: Algebraic Geometry. Springer Science & Business Media, 2013.
Lang, S.: Algebraic Number Theory, Springer Science+Business Media, 1994.
Also check the following pages: