Ideal (ring theory)

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In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3".

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 quotient ring in a similar way as a normal subgroup in group theory can be used to construct a quotient group. The concept of an order ideal in order theory is derived from the notion of ideal in ring theory.

A fractional ideal is a generalization of an ideal, and the usual ideals are sometimes called integral ideals for clarity.



Ideals were first proposed by Richard Dedekind in 1876 in the third edition of his book Vorlesungen über Zahlentheorie (English: Lectures on Number Theory). They were a generalization of the concept of ideal numbers developed by Ernst Kummer [1] [2]. Later the concept was expanded by David Hilbert and especially Emmy Noether.

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