# Ideal (ring theory)

In abstract algebra , an ideal is a subset of a ring that contains the zero element and is closed against the addition and subtraction of elements of the ideal and closed against multiplication with any ring elements. For example, the sum and difference of two even numbers are even again and the product of an even number with any whole number is also even. In addition, the 0 is even as additive neutral. That is, the set of even numbers is an ideal in the ring of whole numbers.

The term “ideal” is derived from the term “ideal number”: ideals can be viewed as generalizations of numbers .

The concept of ideals has its origin in the algebraic number theory of the 19th century by Ernst Eduard Kummer and was further developed by Richard Dedekind and Leopold Kronecker . For David Hilbert , an ideal was a system of infinitely many whole algebraic numbers in a rationality domain ( algebraic number field ), with the property that it also contained all linear combinations of these (with whole algebraic numbers as coefficients). This definition corresponds to today's concept of the broken ideal .

## "Ideal Numbers"

The origin of the ideals lies in the observation that in rings like the uniqueness of the decomposition into irreducible elements it does not apply: So is ${\ displaystyle \ mathbb {Z} \ left [{\ sqrt {-5}} \ right] = \ left \ {a + b \ cdot {\ sqrt {-5}} \ mid a, b \ in \ mathbb { Z} \ right \}}$

${\ displaystyle 6 = 2 \ cdot 3 = \ left (1 + {\ sqrt {-5}} \ right) \ cdot \ left (1 - {\ sqrt {-5}} \ right),}$

and the two factors of any decomposition are irreducible. Ernst Eduard Kummer found that the uniqueness can sometimes be restored by adding more ideal numbers. In the example, the factorizations are obtained by adding the number${\ displaystyle \ mathrm {i}}$

${\ displaystyle 2 = \ left (1+ \ mathrm {i} \ right) (1- \ mathrm {i}), \ quad 3 = {\ frac {1 + {\ sqrt {-5}}} {1+ \ mathrm {i}}} \ cdot {\ frac {1 - {\ sqrt {-5}}} {1- \ mathrm {i}}}}$

( you can see that the fractions on the right side are whole by their norms) as well

${\ displaystyle 1 \ pm {\ sqrt {-5}} = {\ frac {1 \ pm {\ sqrt {-5}}} {1 \ pm \ mathrm {i}}} \ cdot (1 \ pm \ mathrm {i}),}$

and the uniqueness is restored. From today's point of view, the introduction of the ideal number corresponds to the transition to the ( totality ring of) Hilbert's class field , in which all ideals (the totality ring) of an algebraic number field become main ideals . ${\ displaystyle \ mathrm {i}}$

Richard Dedekind realized that one can avoid these ideal numbers by considering the totality of all numbers divisible by them instead. The numbers and in the example have the common ideal prime factor , and the multiples of this number lying in are precisely the prime ideal${\ displaystyle 2}$${\ displaystyle 1 + {\ sqrt {-5}}}$${\ displaystyle 1+ \ mathrm {i}}$${\ displaystyle \ mathbb {Z} [{\ sqrt {-5}}]}$

${\ displaystyle \ left (2.1 + {\ sqrt {-5}} \ right) = \ left \ {a \ cdot 2 + b \ cdot \ left (1 + {\ sqrt {-5}} \ right) \ mid a, b \ in \ mathbb {Z} [{\ sqrt {-5}}] \ right \}.}$

If there is a “real” common factor, then the ideal consists precisely of its multiples and is therefore a main ideal. In wholeness rings of number fields (and more generally in the class of Dedekind rings named after him due to this fact ) one obtains a clear decomposition of every ideal (not equal to zero) into prime ideals (fundamental theorem of ideal theory).

## definition

In order to have suitable terms for non-commutative rings, a distinction is made between left, right and two-sided ideals:

Let it be a subset of a ring . is then called the left ideal if: ${\ displaystyle I}$${\ displaystyle \ mathbf {R} = (R, +, \ cdot)}$${\ displaystyle I}$

1 ,${\ displaystyle 0 \ in I}$
2: for all is ( subgroup criterion ),${\ displaystyle a, b \ in I}$${\ displaystyle a + b \ in I}$
3L: for each and is .${\ displaystyle a \ in I}$${\ displaystyle r \ in R}$${\ displaystyle r \ cdot a \ in I}$

A legal ideal is accordingly if, in addition to 1 and 2 , the following also applies: ${\ displaystyle I}$${\ displaystyle I}$

3R: For each and is .${\ displaystyle a \ in I}$${\ displaystyle r \ in R}$${\ displaystyle a \ cdot r \ in I}$

${\ displaystyle I}$one finally calls the two-sided ideal or only ideal for short if the left and right ideal are fulfilled , i.e. 1, 2, 3L and 3R are fulfilled. ${\ displaystyle I}$

### Remarks

• As an ideal, which contains, it is not empty. In fact, instead of condition 1 because of condition 2 and , the requirement that it is not empty is sufficient.${\ displaystyle I}$${\ displaystyle 0}$${\ displaystyle aa = 0}$${\ displaystyle I}$
• Requirements 1 and 2 are equivalent to the statement that a subgroup is of the additive group .${\ displaystyle (I, +)}$${\ displaystyle (R, +)}$
• Every ideal of also forms a subring of , but generally without a one . In the category of rings with one there is a sub-ring if and only if .${\ displaystyle I}$${\ displaystyle \ mathbf {R}}$${\ displaystyle (I, +, \ cdot)}$${\ displaystyle \ mathbf {R}}$${\ displaystyle 1 \ in \ mathbf {R}}$${\ displaystyle (I, +, \ cdot)}$${\ displaystyle I = \ mathbf {R}}$
• A left as well as a right ideal in is nothing but a - sub-module of , seen as -left- or -Rechtsmodul .${\ displaystyle I}$${\ displaystyle \ mathbf {R}}$${\ displaystyle \ mathbf {R}}$ ${\ displaystyle (I, +)}$${\ displaystyle \ mathbf {R}}$${\ displaystyle \ mathbf {R}}$${\ displaystyle \ mathbf {R}}$${\ displaystyle \ mathbf {R}}$${\ displaystyle (R, +)}$
• If the ring is commutative, then all three terms coincide, but in a non-commutative ring they can be different.

## Examples

• The set of even whole numbers is an ideal in the ring of all whole numbers .${\ displaystyle 2 \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$
• The set of odd integers is not an ideal in ; it does not meet any of the three conditions.${\ displaystyle 2 \ mathbb {Z} +1}$${\ displaystyle \ mathbb {Z}}$
• The set of all polynomials with real coefficients, which are divisible by, form an ideal in the polynomial ring . The body is isomorphic to the complex numbers and is even maximally ideal .${\ displaystyle x ^ {2} +1}$ ${\ displaystyle \ mathbb {R} [X]}$${\ displaystyle \ mathbb {R} [X] / \ left (x ^ {2} +1 \ right)}$${\ displaystyle (x ^ {2} +1)}$
• The ring of all continuous functions from to contains the ideal of the functions with . Another ideal in FIG. 4 is the continuous functions with compact support , i.e. H. all functions that are equal to 0 for sufficiently large arguments.${\ displaystyle C (\ mathbb {R})}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle f}$${\ displaystyle f (1) = 0}$${\ displaystyle C (\ mathbb {R})}$
• The non-commutative ring of the Hurwitz quaternions contains left and right ideals as well as two-sided ideals. However, they are all main ideals.
• The quantities and are always ideals of a ring . This is called the zero ideal and, if R has a one , the one ideal. If and its only two-sided ideals are called simple . A commutative simple ring with one that is not the zero ring is a body .${\ displaystyle \ {0 \}}$${\ displaystyle R}$${\ displaystyle R}$${\ displaystyle \ {0 \} = (0)}$${\ displaystyle 1}$${\ displaystyle R = (1)}$${\ displaystyle \ {0 \}}$${\ displaystyle R}$${\ displaystyle R}$

## Generation of ideals

All left and right ideals and all bilateral ideals each form a system of envelopes . The associated ideal operators are rarely also referred to as. ${\ displaystyle (\;),}$${\ displaystyle \ langle \; \ rangle}$

If a subset of the ring is then called ${\ displaystyle A}$${\ displaystyle R,}$

${\ displaystyle (A): = \ bigcap _ {J \ \ mathrm {Ideal \ von} \ R \ atop \ A \ subseteq J} J}$

the ideal generated by, it is the smallest (left, right or two-sided) ideal in which it contains. ${\ displaystyle A}$ ${\ displaystyle R,}$${\ displaystyle A}$

Has a single element so is ${\ displaystyle R}$${\ displaystyle 1,}$

${\ displaystyle (A) = \ {r_ {1} a_ {1} s_ {1} + \ dotsb + r_ {n} a_ {n} s_ {n} \ mid r_ {i}, s_ {i} \ in R, a_ {i} \ in A \},}$

and if is also commutative, the following even applies: ${\ displaystyle R}$

${\ displaystyle (A) = \ {r_ {1} a_ {1} + \ dotsb + r_ {n} a_ {n} \ mid r_ {i} \ in R, a_ {i} \ in A \}.}$

The main ideal produced by an element is ${\ displaystyle a}$

${\ displaystyle (a): = \ left (\ {a \} \ right.)}$

### Constructions

If a commutative ring with one and an ideal, then the radical of , which is defined as , is also an ideal. ${\ displaystyle R}$${\ displaystyle I \ subseteq R}$ ${\ displaystyle {\ sqrt {I}}}$${\ displaystyle I}$${\ displaystyle {\ sqrt {I}}: = \ {x \ in R \ mid \ exists r \ in \ mathbb {N}: x ^ {r} \ in I \}}$

If there is a ring, then there are two ideals : ${\ displaystyle R}$${\ displaystyle I, J \ subseteq R}$

${\ displaystyle I \ cap J = (I \ cap J)}$
• The set-theoretic union is generally not ideal, but the sum is an ideal:${\ displaystyle I \ cup J}$
${\ displaystyle I + J: = \ {a + b \ mid a \ in I, b \ in J \} = (I \ cup J)}$
Important: sums and associations of ideals are generally different constructs!
• The so-called complex product, which consists of the set of products of elements from with elements from , is generally not ideal either. As a product of and , therefore, the ideal is defined that is generated by:${\ displaystyle IJ,}$${\ displaystyle I}$${\ displaystyle J}$${\ displaystyle I}$${\ displaystyle J}$${\ displaystyle IJ}$
${\ displaystyle I \ cdot J: = \ left (\ left \ {from \ mid a \ in I, b \ in J \ right \} \ right) = (IJ)}$
If there is no risk of confusion with the complex product, then write the ideal product or in short${\ displaystyle I \ cdot J}$${\ displaystyle IJ.}$
• The quotient of and is an ideal that contains all for which the complex product is a subset of :${\ displaystyle I}$${\ displaystyle J}$${\ displaystyle x \ in R}$${\ displaystyle xJ}$${\ displaystyle I}$
${\ displaystyle I: J: = \ {x \ in R \ mid xJ \ subseteq I \}.}$

### Remarks

• The product of two ideals is always contained in their intersection: Are and coprime, so then even equality holds.${\ displaystyle I \ cdot J \ subseteq I \ cap J.}$${\ displaystyle I}$${\ displaystyle J}$${\ displaystyle I + J = R}$
• The ideal quotient is also often written in brackets in the literature: ${\ displaystyle (I: J).}$
• With the links sum and average, the set of all ideals of a ring forms a modular, algebraic association .
• Some important properties of these connections are summarized in Noether's isomorphism theorems .

## Special ideals

An ideal is called real when it is not whole . This is the case for rings with exactly when it is not in . ${\ displaystyle I}$${\ displaystyle R}$${\ displaystyle 1}$${\ displaystyle 1}$${\ displaystyle I}$

A real ideal is a maximum , if there is no greater real ideal, d. i.e., if for every ideal the following applies: ${\ displaystyle M}$${\ displaystyle I}$

${\ displaystyle M \ subseteq I \ subsetneq R \ Rightarrow M = I}$

With the help of Zorn's lemma it can be shown that every true ideal of a ring is contained in a maximal ideal. In particular, every ring with (except for the zero ring ) has a maximum ideal. ${\ displaystyle 1}$${\ displaystyle 1}$

A real ideal is prime , if for all ideals applies: ${\ displaystyle P}$${\ displaystyle I, J}$

${\ displaystyle I \ cdot J \ subseteq P \ Rightarrow I \ subseteq P}$ or ${\ displaystyle J \ subseteq P}$

In a ring with , every maximal ideal is prime. ${\ displaystyle 1}$

## Factor Rings and Cores

Ideals are important because they appear as kernels of ring homomorphisms and allow the definition of factor rings.

A ring homomorphism from the ring into the ring is an image with ${\ displaystyle f}$${\ displaystyle R}$${\ displaystyle S}$${\ displaystyle f \ colon R \ to S}$

${\ displaystyle {\ begin {array} {ll} f (0_ {R}) = 0_ {S}, & f (a + b) = f (a) + f (b), & f (ab) = f (a ) f (b) \ end {array}}}$ for all ${\ displaystyle a, b \ in R.}$

The core of is defined as ${\ displaystyle f}$

${\ displaystyle \ ker (f): = \ {a \ in R \ mid f (a) = 0_ {S} \}.}$

The core is always a two-sided ideal of ${\ displaystyle R.}$

Conversely, if you start with a two-sided ideal of then you can define the factor ring (read: “ modulo ”; not to be confused with a factorial ring ), the elements of which define the form ${\ displaystyle I}$${\ displaystyle R,}$ ${\ displaystyle R / I}$${\ displaystyle R}$${\ displaystyle I}$

${\ displaystyle a + I: = \ {a + i \ mid i \ in I \}}$

for one out . The image ${\ displaystyle a}$${\ displaystyle R}$

${\ displaystyle p \ colon R \ to R / I, \, a \ mapsto a + I}$

is a surjective ring homomorphism, the core of which is precisely the ideal . Thus the ideals of a ring are exactly the kernels of ring homomorphisms of${\ displaystyle I}$${\ displaystyle R}$${\ displaystyle R.}$

If the ring is commutative and a prime ideal, then it is an integrity ring , is a maximum ideal, then there is even a body . ${\ displaystyle R}$${\ displaystyle P}$${\ displaystyle R / P}$${\ displaystyle M}$${\ displaystyle R / M}$

The extreme examples of factor rings of a ring arise by splitting out the ideals or the factor ring is isomorphic to and is the trivial ring${\ displaystyle R}$${\ displaystyle (0)}$${\ displaystyle R.}$${\ displaystyle R / (0)}$${\ displaystyle R,}$${\ displaystyle R / R}$${\ displaystyle \ {0 \}.}$

## Norm of an ideal

For total rings of a number field , a norm of a (whole) ideal can be defined by (and for the zero ideal ). This norm is always a finite number and is related to the norm of the body extension for main ideals . In addition, this norm is multiplicative, i. H. . More generally, these norms are also considered for ideals in orders in number fields. ${\ displaystyle A}$ ${\ displaystyle K}$${\ displaystyle I}$${\ displaystyle N (I): = \ mathrm {card} (A / I))}$${\ displaystyle N ((0)): = 0}$ ${\ displaystyle N_ {K / \ mathbb {Q}},}$${\ displaystyle (a)}$${\ displaystyle | N_ {K / \ mathbb {Q}} (a) | = N ((a)).}$${\ displaystyle N (I \ cdot J)) = N (I) N (J)}$

## Individual evidence

1. Felix Klein: Lectures on the development of mathematics in the 19th century. Part 1. Springer, Berlin 1926 ( The basic teachings of the mathematical sciences in individual representations. 24, ), Chapter VII, section Theory of algebraic integers ... p. 321 f.
2. Felix Klein: Lectures on the development of mathematics in the 19th century. Part 1. Springer, Berlin 1926 ( The basic teachings of the mathematical sciences in individual representations. 24, ), p. 323.
3. J. Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 ; Theorem I.3.3.
4. Lecture Algebra I. (PDF; 493 kB) Retrieved on August 24, 2013 .

## literature

• Felix Klein : Lectures on the development of mathematics in the 19th century. Part 1. Springer, Berlin 1926 ( The basic teachings of the mathematical sciences in individual representations. 24, ).
• Ernst Eduard Kummer : On the decomposition of the complex numbers formed from the roots of the unit into their prime factors. In: Journal for pure and applied mathematics . 35, 1847, pp. 327-367.
• David Hilbert : number report "The theory of algebraic number fields, annual report of the German Mathematicians Association", Vol. 4 pp. 175–546 1897 [1]