From Wikipedia, the free encyclopedia
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 ,
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
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.
External
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