SQUARE45

Definition

(For the sake of brevity, some results are stated only for left ideals but are usually also true for right ideals with appropriate notation changes.) - In a ring R, the set R itself forms a two-sided ideal of R called the unit ideal. It is often also denoted by

(1)

since it is precisely the two-sided ideal generated (see below) by the unity ⁠

1_{R}

⁠. Also, the set

\{0_{R}\}

consisting of only the additive identity 0_{R} forms a two-sided ideal called the zero ideal and is denoted by ⁠

(0)

⁠. Every (left, right or two-sided) ideal contains the zero ideal and is contained in the unit ideal. - An (left, right or two-sided) ideal that is not the unit ideal is called a proper ideal (as it is a proper subset). Note: a left ideal

{\mathfrak {a}}

is proper if and only if it does not contain a unit element, since if

u\in {\mathfrak {a}}

is a unit element, then

r=(ru^{-1})u\in {\mathfrak {a}}

for every ⁠

r\in R

⁠. Typically there are plenty of proper ideals. In fact, if R is a skew-field, then

(0),(1)

are its only ideals and conversely: that is, a nonzero ring R is a skew-field if

(0),(1)

are the only left (or right) ideals. (Proof: if

x

is a nonzero element, then the principal left ideal

Rx

(see below) is nonzero and thus

Rx=(1)

; i.e.,

yx=1

for some nonzero ⁠

y

⁠. Likewise,

zy=1

for some nonzero

z

. Then

z=z(yx)=(zy)x=x

.) - The even integers form an ideal in the ring

\mathbb {Z}

of all integers, since the sum of any two even integers is even, and the product of any integer with an even integer is also even; this ideal is usually denoted by ⁠

2\mathbb {Z}

⁠. More generally, the set of all integers divisible by a fixed integer

n

is an ideal denoted ⁠

n\mathbb {Z}

⁠. In fact, every non-zero ideal of the ring

\mathbb {Z}

is generated by its smallest positive element, as a consequence of Euclidean division, so

\mathbb {Z}

is a principal ideal domain. - The set of all polynomials with real coefficients that are divisible by the polynomial

x^{2}+1

is an ideal in the ring of all real-coefficient polynomials ⁠

\mathbb {R} [x]

⁠. - Take a ring

R

and positive integer ⁠

n

⁠. For each ⁠

1\leq i\leq n

⁠, the set of all

n\times n

matrices with entries in

R

whose

i

-th row is zero is a right ideal in the ring

M_{n}(R)

of all

n\times n

matrices with entries in ⁠

R

⁠. It is not a left ideal. Similarly, for each ⁠

1\leq j\leq n

⁠, the set of all

n\times n

matrices whose

j

-th column is zero is a left ideal but not a right ideal. - The ring

C(\mathbb {R} )

of all continuous functions

f

from

\mathbb {R}

to

\mathbb {R}

under pointwise multiplication contains the ideal of all continuous functions

f

such that ⁠

f(1)=0

⁠. Another ideal in

C(\mathbb {R} )

is given by those functions that 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 ⁠

\vert x\vert >L

⁠. - A ring is called a simple ring if it is nonzero and has no two-sided ideals other than ⁠

(0),(1)

⁠. Thus, a skew-field is simple and a simple commutative ring is a field. The matrix ring over a skew-field is a simple ring. - If

f:R\to S

is a ring homomorphism, then the kernel

\ker(f)=f^{-1}(0_{S})

is a two-sided ideal of ⁠

R

⁠. By definition, ⁠

f(1_{R})=1_{S}

⁠, and thus if

S

is not the zero ring (so ⁠

1_{S}\neq 0_{S}

⁠), then

\ker(f)

is a proper ideal. More generally, for each left ideal I of S, the pre-image

f^{-1}(I)

is a left ideal. If I is a left ideal of R, then

f(I)

is a left ideal of the subring

f(R)

of S: unless f is surjective,

f(I)

need not be an ideal of S; see also § Extension and contraction of an ideal. - Ideal correspondence: Given a surjective ring homomorphism ⁠

f:R\to S

⁠, there is a bijective order-preserving correspondence between the left (resp. right, two-sided) ideals of

R

containing the kernel of

f

and the left (resp. right, two-sided) ideals of

S

: the correspondence is given by

I\mapsto f(I)

and the pre-image ⁠

J\mapsto f^{-1}(J)

⁠. Moreover, for commutative rings, this bijective correspondence restricts to prime ideals, maximal ideals, and radical ideals (see the Types of ideals section for the definitions of these ideals). - If M is a left R-module and

S\subset M

a subset, then the annihilator

\operatorname {Ann} _{R}(S)=\{r\in R\mid rs=0,s\in S\}

of S is a left ideal. Given ideals

{\mathfrak {a}},{\mathfrak {b}}

of a commutative ring R, the R-annihilator of

({\mathfrak {b}}+{\mathfrak {a}})/{\mathfrak {a}}

is an ideal of R called the ideal quotient of

{\mathfrak {a}}

by

{\mathfrak {b}}

and is denoted by ⁠

({\mathfrak {a}}:{\mathfrak {b}})

⁠; it is an instance of idealizer in commutative algebra. - Let

{\mathfrak {a}}_{i},i\in S

be an ascending chain of left ideals in a ring R; i.e.,

S

is a totally ordered set and

{\mathfrak {a}}_{i}\subset {\mathfrak {a}}_{j}

for each ⁠

i<j

⁠. Then the union

\textstyle \bigcup _{i\in S}{\mathfrak {a}}_{i}

is a left ideal of R. (Note: this fact remains true even if R is without the unity 1.) - The above fact together with Zorn's lemma proves the following: if

E\subset R

is a possibly empty subset and

{\mathfrak {a}}_{0}\subset R

is a left ideal that is disjoint from E, then there is an ideal that is maximal among the ideals containing

{\mathfrak {a}}_{0}

and disjoint from E. (Again this is still valid if the ring R lacks the unity 1.) When

R\neq 0

, taking

{\mathfrak {a}}_{0}=(0)

and ⁠

E=\{1\}

⁠, in particular, there exists a left ideal that is maximal among proper left ideals (often simply called a maximal left ideal); see Krull's theorem for more. - A left (resp. right, two-sided) ideal generated by a single element x is called the principal left (resp. right, two-sided) ideal generated by x and is denoted by

Rx

(resp. ⁠

xR,RxR

⁠). The principal two-sided ideal

RxR

is often also denoted by ⁠

(x)

⁠ or

\langle x\rangle

. - An arbitrary union of ideals need not be an ideal, but the following is still true: given a possibly empty subset X of R, there is the smallest left ideal containing X, called the left ideal generated by X and is denoted by ⁠

RX

⁠. Such an ideal exists since it is the intersection of all left ideals containing X. Equivalently,

RX

is the set of all the (finite) left R-linear combinations of elements of X over R:

RX=\{r_{1}x_{1}+\dots +r_{n}x_{n}\mid n\in \mathbb {N} ,r_{i}\in R,x_{i}\in X\}

(since such a span is the smallest left ideal containing X.) A right (resp. two-sided) ideal generated by X is defined in the similar way. For "two-sided", one has to use linear combinations from both sides; i.e.,

RXR=\{r_{1}x_{1}s_{1}+\dots +r_{n}x_{n}s_{n}\mid n\in \mathbb {N} ,r_{i}\in R,s_{i}\in R,x_{i}\in X\}.

If

X=\{x_{1},\dots ,x_{n}\}

is a finite set, then

RXR

is also written as ⁠

(x_{1},\dots ,x_{n})

⁠ or

\langle x_{1},...,x_{n}\rangle

. More generally, the two-sided ideal generated by a (finite or infinite) set of indexed ring elements

X=\{x_{i}\}_{i\in I}

is denoted

(X)=(x_{i})_{i\in I}

or

\langle X\rangle =\langle x_{i}\rangle _{i\in I}

. - There is a bijective correspondence between ideals and congruence relations (equivalence relations that respect the ring structure) on the ring: Given an ideal

I

of a ring ⁠

R

⁠, let

x\sim y

if ⁠

x-y\in I

⁠. Then

\sim

is a congruence relation on ⁠

R

⁠. Conversely, given a congruence relation

\sim

on ⁠

R

⁠, let ⁠

I=\{x\in R:x\sim 0\}

⁠. Then

I

is an ideal of ⁠

R

⁠. Cite error: There are tags on this page, but the references will not show without a {{reflist|group=note}} template (see the help page). - ^ ^{a}^{b}^{c}Dummit & Foote (2004), p. 243. - ^Lang (2005), Section III.2. - ^Dummit & Foote (2004), p. 244.

Theorem

Dirichlet's Unit Theorem

Advanced

Determines the rank of the group of units in the ring of integers of a number field.

Ideal Theory

Sequence of Expressions

Definitions

Examples and properties

Dirichlet's Unit Theorem