Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars Both algebraic geometry and algebraic number theory build on commutative algebra. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Mathematics, algebraic number theory is a major branch of Number theory which studies the Algebraic structures related to Algebraic integers Prominent examples of commutative rings include polynomial rings, rings of algebraic integers, including the ordinary integers Z, and p-adic integers. In Mathematics, especially in the field of Abstract algebra, a polynomial ring is a ring formed from the set of Polynomials in one or more variables This article deals with the ring of complex numbers integral over Z. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897

Commutative algebra is the main technical tool in the local study of schemes. In Mathematics, a scheme is an important concept connecting the fields of Algebraic geometry, Commutative algebra and Number theory.

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. In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the

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. In Mathematics, ideal theory is the theory of ideals in Commutative rings and is the precursor name for the contemporary subject of Commutative Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important Ernst Eduard Kummer ( 29 January 1810 - 14 May 1893) was a German Mathematician. Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued Later, David Hilbert introduced the term ring to generalize the earlier term number ring. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most 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. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex Invariant theory is a branch of Abstract algebra that studies actions of groups on algebraic varieties from the point of view of their effect In turn, Hilbert strongly influenced Emmy Noether, to whom we owe much of the abstract and axiomatic approach to the subject. Amalie Emmy Noether, ˈnøːtɐ (23 March 1882 – 14 April 1935 was a German Mathematician known for her groundbreaking contributions to Abstract algebra and 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. For other persons named Lasker see Lasker#People with the surname Lasker. In Mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be written as an intersection In Mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be written as an intersection

Much of the modern development of commutative algebra emphasizes modules. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars 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. Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued Amalie Emmy Noether, ˈnøːtɐ (23 March 1882 – 14 April 1935 was a German Mathematician known for her groundbreaking contributions to Abstract algebra and

See also

• List of commutative algebra topics
• Combinatorial commutative algebra
• Gröbner basis
• Homological algebra

References

• Michael Atiyah & Ian G. MacDonald, Introduction to Commutative Algebra, Massachusetts : Addison-Wesley Publishing, 1969. This is a list of Commutative algebra topics, by Wikipedia page Combinatorial commutative algebra is a relatively new rapidly developing mathematical discipline In Computer algebra, computational Algebraic geometry, and computational Commutative algebra, a Gröbner basis is a particular kind of generating subset Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting Sir Michael Francis Atiyah, OM, FRS, FRSE (b April 22, 1929) is a British Mathematician, and one of the Ian G Macdonald (born 1928 in London, England) is a British mathematician known for his contributions to Symmetric functions Special functions Introduction to Commutative Algebra is a well-known Commutative algebra textbook written by Michael Atiyah and Ian G
• Bourbaki, Nicolas, Commutative algebra. Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition 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. Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition 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. David Eisenbud (born 8 April, 1947) is an American Mathematician.
• 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. Masayoshi Nagata ( Japanese: 永田 雅宜 Nagata Masayoshi; born 1927 in Aichi Japan (February 9 1927–August 27 2008 is a Japanese mathematician 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. Oscar Zariski (born Oscher Zaritsky 24 April 1899 in Kobrin, Poland (today Belarus) died 4 July Pierre Samuel (born 12 September 1921 in Paris) is a French mathematician known for his work in Commutative algebra and its applications 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 links

• List of Commutative Algebraists
• The Commutative Algebra Community