Sajun.org
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.
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 additional
module concept, present in some form in Kronecker's work, is technically an improvement on working always directly on the special case of ''ideals''. Its adoption is attributed to
Emmy Noether's influence.
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.
For links, see
list of commutative algebra topics.
de:Kommutative Algebra
it:algebra commutativa
