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For instance, in rings one studies prime ideals instead of prime numbers, one defines coprime ideals as a generalization of coprime numbers, and one can prove a generalized Chinese remainder theorem about ideals. In a certain class of rings important in number theory, the Dedekind domains, one can even recover a version of the fundamental theorem of arithmetic: in these rings, every nonzero ideal can be uniquely written as a product of prime ideals.
An ideal can be used to construct a factor ring in a similar way as a normal subgroup in group theory can be used to construct a factor group. The concept of an order ideal in order theoryOrder theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. This article gives a detailed introduction to the field and includes some of the most basic definitions. is derived from the notion of ideal in ring theory.
Ideals were first proposed by Dedekind in 1876Events January events January 31 The United States orders all Native Americans to move into reservations. February events February 2 The National League of Professional Baseball Clubs of Major League Baseball is formed. February 14 Alexander Graham Bell a in the third edition of his book Vorlesungen über ZahlentheorieVorlesungen uber Zahlentheorie Lectures on Number Theory is a textbook of number theory written by German mathematicians P. Dirichlet and Richard Dedekind, and published in 1863. Based on Dirichlet's number theory course at the University of Gottingen, th (engl.: Lectures on number theory). They were a generalization of the concept of ideal numberIn mathematics an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Kummer while trying to solve Fermat's last theorem, and lead to Dedekind's definition of ideals for ringss developed by Ernst KummerErnst Eduard Kummer ( 29 January 1810 in Sorau, Brandenburg, Prussia 14 May 1893 in Berlin, Germany) was a German mathematician. Capable in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a Gym. Later the concept was expanded by David HilbertDavid Hilbert ( January 23, 1862 February 14, 1943) was a German mathematician born in Wehlau, near Konigsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th cen and especially Emmy Noether.
Let R be a ring and with (R,+) the abelian group of the ring. Then a subset I of R is called right ideal if
and left ideal if
When R is commutative ring the notion of left ideal and right ideal coincide and the two-sided ideal is simply called ideal. To keep the following definitions shorter we will only consider commutative rings .
We call I a proper ideal if it is a real subset of R. A proper ideal I is a called maximal ideal if there exists no other ideal J (the trivial ideal R excluded) with I a subset of J. A proper ideal I is called a prime ideal if for all ab in I it follows either a or b in I.
If we can write every element x of I as
where ik is an element of I and rk is an element of R we say I is finitely generated. If I is generated by only one element we call I a principal ideal.
If A is any subset of the ring R, then we can define the ideal generated by A to be the smallest ideal of R containing A; it is denoted by <A> or (A) and contains all finite sums of the form
with each ri and si in R and each ai in A. The principal ideals mentioned above are the special case when A is just the singleton {a}.
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