Commutative algebra
From Free net encyclopedia
In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. It is foundational both for algebraic geometry and for algebraic number theory. The most prominent example for commutative rings are polynomial rings.
Given the scheme concept, commutative algebra is reasonably thought of as either the local theory or the affine theory of algebraic geometry.
The general study of rings that are not required to be commutative is known as noncommutative algebra; it is pursued in ring theory, representation theory and in other areas such as Banach algebra theory.
History
The subject's real founder, in the days when it was called ideal theory, should be considered to be David Hilbert. He seems to have thought of it (around 1900) as an alternate approach that could replace the then-fashionable complex function theory. In line with his thinking, computational aspects were secondary to the structural.
The modern development of commutative algebra places additional emphasis on concept of a module, present already in some form in Kronecker's work. Modules present a technical improvement on working always directly on the special case of ideals. Now an ideal a in a ring R and its quotient ring R/a can be treated on equal footing as both are special cases of a module. The adoption of the module concept is attributed to Emmy Noether's influence.
See also
fr:Algèbre commutative it:Algebra commutativa ru:Коммутативная алгебра