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Encyclopedia > Commutative algebra

In abstract algebra, commutative algebra 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. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... 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 to lie in a field, the scalars may lie in an arbitrary ring. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... This article or section does not cite its references or sources. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In mathematics, an algebraic integer is a complex number &#945; that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ... The integers are commonly denoted by the above symbol. ... 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, there is a close relationship between spaces, which are geometric in nature, and the numerical functions on them. ... In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ...

## Contents

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 (also a world chess champion), who introduced primary ideals and proved the first version of the Lasker–Noether theorem. In mathematics, ideal theory is the theory of ideals in commutative rings; and is the precursor name for the contemporary subject of commutative algebra. ... Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 â€“ February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ... In mathematics, the term ideal has multiple meanings. ... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ... David Hilbert (January 23, 1862, KÃ¶nigsberg, East Prussia â€“ February 14, 1943, GÃ¶ttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... In mathematics, invariant theory refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations. ... Amalie Emmy Noether [1] (March 23, 1882 â€“ April 14, 1935) was a German-born mathematician, said by Einstein in eulogy to be [i]n the judgment of the most competent living mathematicians, [...] the most significant creative mathematical genius thus far produced since the higher education of women began. ... Emanuel Lasker (December 24, 1868 â€“ January 11, 1941) was a German World Chess Champion, mathematician, and philosopher born at Berlinchen in Brandenburg (now Barlinek in Poland). ... The 1984 World Chess Championship was between Anatoly Karpov (left) and Garry Kasparov (right). ... In mathematics, an ideal in a commutative ring is a primary ideal if for all elements , we have that if , then either or for some This is clearly a generalization of the notion of a prime ideal, and (very) loosely mirrors the relationship in between prime numbers and prime powers. ... In mathematics, the Laskerâ€“Noether theorem provides a vast generalization of the fundamental theorem of arithmetic to embrace the rings of algebraic geometry. ...

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. 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 to lie in a field, the scalars may lie in an arbitrary ring. ... Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the natural numbers; all else is the work of man (Bell 1986, p. ... Amalie Emmy Noether [1] (March 23, 1882 â€“ April 14, 1935) was a German-born mathematician, said by Einstein in eulogy to be [i]n the judgment of the most competent living mathematicians, [...] the most significant creative mathematical genius thus far produced since the higher education of women began. ...

This is a list of commutative algebra topics, by Wikipedia page. ...

## References

• Michael Atiyah & Ian G. MacDonald, Introduction to Commutative Algebra, Massachusetts : Addison-Wesley Publishing, 1969.
• David Eisenbud, Commutative Algebra With a View Toward Algebraic Geometry, New York : Springer-Verlag, 1999.
• Hideyuki Matsumura, translated by Miles Reid, Commutative Ring Theory (Cambridge Studies in Advanced Mathematics),Cambridge, UK : Cambridge University Press, 1989.
• Miles Reid, Undergraduate Commutative Algebra (London Mathematical Society Student Texts), Cambridge, UK : Cambridge University Press, 1996.
• Jean-Pierre Serre, Algèbre locale, multiplicités

Sir Michael Francis Atiyah, OM, FRS (born 22 April 1929) is a mathematician who was born in London. ... Ian G. Macdonald (born 1928) is a British mathematician known for his contributions to symmetric functions, special functions, Lie algebra theory and other aspects of algebraic combinatorics. ... Introduction to Commutative Algebra is a commutative algebra textbook written by M. F. Atiyah and I. G. Macdonald. ... David Eisenbud (born 8 April 1947) is an American mathematician. ... Miles Reid (born 30 January 1948, Hoddesdon, England) is a mathematician at the University of Warwick who works in algebraic geometry. ... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ...

Results from FactBites:

 13: Commutative rings and algebras (2760 words) Of particular interest are several classes of rings of interest in number theory, field theory, algebraic geometry, and related areas; however, other classes of rings arise, and a rich structure theory arises to analyze commutative rings in general, using the concepts of ideals, localizations, and homological algebra. In the case of geometric algebras, this is the setting for families of subvarieties, including nested and intersecting ones. Typically one classifies problems as Algebraic Geometry when stated in terms of points, hypersurfaces, divisors, and other geometric objects, and as Commutative Algebra when stated in terms of ideals and coordinate rings, although in practice techniques from both areas are used in tandem.
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