Commutative algebra

This page deals with the area of study. For algebras which are commutative, see algebra (disambiguation).

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 $\mathbb{Z}$, and p-adic integers.

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

The study of rings which are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras.

History

The subject, first known as ideal theory, began with Richard Dedekind's work on ideals, itself based on the earlier work of Ernst Kummer and Leopold Kronecker. Later, David Hilbert introduced the term ring to generalize the earlier term number ring. Hilbert introduced a more abstract approach to replace the more concrete and computationally oriented methods grounded in such things as complex analysis and classical invariant theory. In turn, Hilbert strongly influenced Emmy Noether, to whom we owe much of the abstract and axiomatic approach to the subject. Another important milestone was the work of Hilbert's student Emanuel Lasker, who introduced primary ideals and proved the first version of the Lasker–Noether theorem.

Much of the modern development of commutative algebra emphasizes modules. Both ideals of a ring R and R-algebras are special cases of R-modules, so module theory encompasses both ideal theory and the theory of ring extensions. Though it was already incipient in Kronecker's work, the modern approach to commutative algebra using module theory is usually credited to Emmy Noether.

See also

Main article: Outline of commutative algebra

• Combinatorial commutative algebra
• Gröbner basis
• Homological algebra

References

• Michael Atiyah & Ian G. Macdonald, Introduction to Commutative Algebra, Massachusetts : Addison-Wesley Publishing, 1969.
• Bourbaki, Nicolas, Commutative algebra. Chapters 1--7. Translated from the French. Reprint of the 1989 English translation. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1998. xxiv+625 pp. ISBN 3-540-64239-0
• Bourbaki, Nicolas, Éléments de mathématique. Algèbre commutative. Chapitres 8 et 9. (Elements of mathematics. Commutative algebra. Chapters 8 and 9) Reprint of the 1983 original. Springer, Berlin, 2006. ii+200 pp. ISBN 978-3-540-33942-7
• David Eisenbud, Commutative Algebra With a View Toward Algebraic Geometry, New York : Springer-Verlag, 1999.
• Rémi Goblot, "Algèbre commutative, cours et exercices corrigés", 2e édition, Dunod 2001, ISBN 2-10-005779-0
• Ernst Kunz, "Introduction to Commutative algebra and algebraic geometry", Birkhauser 1985, ISBN 0-8176-3065-1
• Matsumura, Hideyuki, Commutative algebra. Second edition. Mathematics Lecture Note Series, 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. xv+313 pp. ISBN 0-8053-7026-9
• Matsumura, Hideyuki, Commutative Ring Theory. Second edition. Translated from the Japanese. Cambridge Studies in Advanced Mathematics, Cambridge, UK : Cambridge University Press, 1989. ISBN 0-521-36764-6
• Nagata, Masayoshi, Local rings. Interscience Tracts in Pure and Applied Mathematics, No. 13. Interscience Publishers a division of John Wiley and Sons, New York-London 1962 xiii+234 pp.
• Miles Reid, Undergraduate Commutative Algebra (London Mathematical Society Student Texts), Cambridge, UK : Cambridge University Press, 1996.
• Jean-Pierre Serre, Local algebra. Translated from the French by CheeWhye Chin and revised by the author. (Original title: Algèbre locale, multiplicités) Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2000. xiv+128 pp. ISBN 3-540-66641-9
• Sharp, R. Y., Steps in commutative algebra. Second edition. London Mathematical Society Student Texts, 51. Cambridge University Press, Cambridge, 2000. xii+355 pp. ISBN 0-521-64623-5
• Zariski, Oscar; Samuel, Pierre, Commutative algebra. Vol. 1, 2. With the cooperation of I. S. Cohen. Corrected reprinting of the 1958, 1960 edition. Graduate Texts in Mathematics, No. 28, 29. Springer-Verlag, New York-Heidelberg-Berlin, 1975.