Commutative Algebra, Algebraic Geometry, and Algebraic Number Theory

A Brief About Commutative Algebra

Commutative algebra is the 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 include polynomial rings, rings of algebraic integers, including the ordinary integers Z, and p-adic integers.

Commutative algebra is the main technical tool in the local study of schemes.

Introductory Books on Commutative Algebra

  • Reid, Miles. Undergraduate commutative algebra. Vol. 29. Cambridge University Press, 1995.
  • Sharp, Rodney Y., ed. Steps in commutative algebra. Vol. 51. Cambridge university press, 2000.
  • Watkins, John J. Topics in commutative ring theory. Princeton University Press, 2009.

Some Books on Commutative Algebra for Further Studies

  • Atiyah, M., and I. G. Macdonald. "Commutative algebra." Addison-Wesley, Reading, Mass (1969).
  • Winfried Bruns, and H. Jürgen Herzog. Cohen-Macaulay rings. Cambridge University Press, 1998.
  • Bruns, Winfried, and Udo Vetter. Determinantal rings. Springer, 1988.
  • Eisenbud, David. Commutative Algebra: with a view toward algebraic geometry. Vol. 150. Springer Science & Business Media, 1995.
  • Gillman, Leonard, and Meyer Jerison. Rings of continuous functions. Princeton, NJ, 1960.
  • Gilmer, Robert W. Multiplicative ideal theory. Vol. 12. M. Dekker, 1972.
  • Kaplansky, Irving. Commutative rings. Boston, 1970.
  • Kreuzer, Martin, and Lorenzo Robbiano. Computational commutative algebra 2. Vol. 2. Springer Science & Business Media, 2005.
  • Larsen, Max D., and Paul Joseph McCarthy. Multiplicative theory of ideals. Vol. 43. Academic press, 1971.
  • Matsumura, Hideyuki. "Commutative Algebra, (Mathematics Lecture Note Series, 56)." (1980).
  • Matsumura, Hideyuki. Commutative ring theory. Vol. 8. Cambridge university press, 1989.
  • Miller, Ezra, and Bernd Sturmfels. Combinatorial commutative algebra. Vol. 227. Springer Science & Business Media, 2005.
  • Stanley, Richard P. Combinatorics and commutative algebra. Vol. 41. Springer Science & Business Media, 2007.
  • Zariski, Oscar and Pierre Samuel. Commutative algebra II. Vol. 2. Springer Science & Business Media, 1976.

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