# Ideal (ring theory)

(Redirected from Maximal ideal)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept generalizes in an appropriate way 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 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 theory is derived from the notion of ideal in ring theory.

## History

Ideals were first proposed by Dedekind in 1876 in the third edition of his book Vorlesungen über Zahlentheorie (Engl.: Lectures on number theory). They were a generalization of the concept of ideal numbers developed by Ernst Kummer. Later the concept was expanded by David Hilbert and especially Emmy Noether.

## Definitions

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

• (I,+) is a subgroup of (R,+)
• for all i in I and all r in R : i r is still in I

and left ideal if

• (I,+) is a subgroup of (R,+)
• for all i in I and all r in R : r i is still in I

When R is a 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 proper subset of R, that is, I does not equal R. A proper ideal I is called a maximal ideal if there exists no other proper ideal J with I a subset of J. A proper ideal I is called a prime ideal if for all ab in I, either a or b is in I.

If we can write every element x of I as $x=\sum _{{k=0}}^{{n}}r_{{k}}i_{{k}}$

where ik is an element of I and {rk: k=1,...,n} is a fixed finite subset 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

r1a1s1 + ··· + rnansn

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}.

## Examples

• The even integers form an ideal in the ring Z of all integers; it is usually denoted by 2Z. This because the sum of any even integers is even, and the product of any integer with an even integer is also even.
• In the ring Z of integers, every ideal can be generated by a single number (so Z is a principal ideal domain), and the ideal determines the number up to its sign.The concepts of "ideal" and "number" are therefore almost identical in Z (and in any principal ideal domain).
• The set of all polynomials with real coefficients which are divisible by the polynomial x2 + 1 is an ideal in the ring of all polynomials.
• The set of all n-by-n matrices whose last column is zero forms a left ideal in the ring of all n-by-n matrices. It is not a right ideal. The set of all n-by-n matrices whose last row is zero forms a right ideal but not a left ideal.
• The ring C(R) of all continuous functions f from R to R contains the ideal of all continuous functions f such that f(1) = 0. Another ideal in C(R) is given by those functions which vanish for large enough arguments, i.e. those continuous functions f for which there exists a number L > 0 such that f(x) = 0 whenever |x| > L.
• {0} and R are ideals in every ring R. If R is commutative, then R is a field iff it has precisely two ideals, {0} and R.