Everything about Commutative Algebra totally explained
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.
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.
Further Information
Get more info on 'Commutative Algebra'.
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