Editing Ideal (ring theory) (section)

== Examples and properties ==
(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 <math>(1)</math> since it is precisely the two-sided ideal generated (see below) by the unity {{tmath|1= 1_R }}. Also, the set <math>\{ 0_R \}</math> consisting of only the additive identity 0<sub>''R''</sub> forms a two-sided ideal called the '''zero ideal''' and is denoted by {{tmath|1= (0) }}.<ref group=note>Some authors call the zero and unit ideals of a ring ''R'' the '''trivial ideals''' of ''R''.</ref> Every (left, right or two-sided) ideal contains the zero ideal and is contained in the unit ideal.{{sfnp|Dummit|Foote|2004|p=243}}
* An (left, right or two-sided) ideal that is not the unit ideal is called a '''proper ideal''' (as it is a [[proper subset]]).{{sfnp|Lang|2005|loc=Section III.2}} Note: a left ideal <math>\mathfrak{a}</math> is proper if and only if it does not contain a unit element, since if <math>u \in \mathfrak{a}</math> is a unit element, then <math>r = (r u^{-1}) u \in \mathfrak{a}</math> for every {{tmath|1= r \in R }}. Typically there are plenty of proper ideals. In fact, if ''R'' is a [[skew-field]], then <math>(0), (1)</math> are its only ideals and conversely: that is, a nonzero ring ''R'' is a skew-field if <math>(0), (1)</math> are the only left (or right) ideals. (Proof: if <math>x</math> is a nonzero element, then the principal left ideal <math>Rx</math> (see below) is nonzero and thus <math>Rx = (1)</math>; i.e., <math>yx = 1</math> for some nonzero {{tmath|1= y }}. Likewise, <math>zy = 1</math> for some nonzero <math>z</math>. Then <math>z = z(yx) = (zy)x = x</math>.)
* The even [[integer]]s form an ideal in the ring <math>\mathbb{Z}</math> 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 {{tmath|1= 2\mathbb{Z} }}. More generally, the set of all integers divisible by a fixed integer <math>n</math> is an ideal denoted {{tmath|1= n\mathbb{Z} }}. In fact, every non-zero ideal of the ring <math>\mathbb{Z}</math> is generated by its smallest positive element, as a consequence of [[Euclidean division]], so <math>\mathbb{Z}</math> is a [[principal ideal domain]].{{sfnp|Dummit|Foote|2004|p=243}}
* The set of all [[polynomial]]s with real coefficients that are divisible by the polynomial <math>x^2+1</math> is an ideal in the ring of all real-coefficient polynomials {{tmath|1= \mathbb{R}[x] }}.
* Take a ring <math>R</math> and positive integer {{tmath|1= n }}. For each {{tmath|1= 1\leq i\leq n }}, the set of all <math>n\times n</math> [[matrix (mathematics)|matrices]] with entries in <math>R</math> whose <math>i</math>-th row is zero is a right ideal in the ring <math>M_n(R)</math> of all <math>n\times n</math> matrices with entries in {{tmath|1= R }}. It is not a left ideal. Similarly, for each {{tmath|1= 1\leq j\leq n }}, the set of all <math>n\times n</math> matrices whose <math>j</math>-th ''column'' is zero is a left ideal but not a right ideal.
* The ring <math>C(\mathbb{R})</math> of all [[continuous function]]s <math>f</math> from <math>\mathbb{R}</math> to <math>\mathbb{R}</math> under [[pointwise multiplication]] contains the ideal of all continuous functions <math>f</math> such that {{tmath|1= f(1)=0 }}.{{sfnp|Dummit|Foote|2004|p=244}} Another ideal in <math>C(\mathbb{R})</math> is given by those functions that vanish for large enough arguments, i.e. those continuous functions <math>f</math> for which there exists a number <math>L>0</math> such that <math>f(x)=0</math> whenever {{tmath|1= \vert x \vert >L }}.
* A ring is called a [[simple ring]] if it is nonzero and has no two-sided ideals other than {{tmath|1= (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 <math>f: R \to S</math> is a [[ring homomorphism]], then the kernel <math>\ker(f) = f^{-1}(0_S)</math> is a two-sided ideal of {{tmath|1= R }}.{{sfnp|Dummit|Foote|2004|p=243}} By definition, {{tmath|1= f(1_R) = 1_S }}, and thus if <math>S</math> is not the zero ring (so {{tmath|1= 1_S\ne0_S }}), then <math>\ker(f)</math> is a proper ideal. More generally, for each left ideal ''I'' of ''S'', the pre-image <math>f^{-1}(I)</math> is a left ideal. If ''I'' is a left ideal of ''R'', then <math>f(I)</math> is a left ideal of the subring <math>f(R)</math> of ''S'': unless ''f'' is surjective, <math>f(I)</math> need not be an ideal of ''S''; see also {{slink|#Extension and contraction of an ideal}}.
* '''Ideal correspondence''': Given a surjective ring homomorphism {{tmath|1= f: R \to S }}, there is a bijective order-preserving correspondence between the left (resp. right, two-sided) ideals of <math>R</math> containing the kernel of <math>f</math> and the left (resp. right, two-sided) ideals of <math>S</math>: the correspondence is given by <math>I \mapsto f(I)</math> and the pre-image {{tmath|1= 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|Types of ideals]] section for the definitions of these ideals).
* If ''M'' is a left ''R''-[[Module (mathematics)|module]] and <math>S \subset M</math> a subset, then the [[annihilator (ring theory)|annihilator]] <math>\operatorname{Ann}_R(S) = \{ r \in R \mid rs = 0, s \in S \}</math> of ''S'' is a left ideal. Given ideals <math>\mathfrak{a}, \mathfrak{b}</math> of a commutative ring ''R'', the ''R''-annihilator of <math>(\mathfrak{b} + \mathfrak{a})/\mathfrak{a}</math> is an ideal of ''R'' called the [[ideal quotient]] of <math>\mathfrak{a}</math> by <math>\mathfrak{b}</math> and is denoted by {{tmath|1= (\mathfrak{a} : \mathfrak{b}) }}; it is an instance of [[idealizer]] in commutative algebra.
* Let <math>\mathfrak{a}_i, i \in S</math> be an '''[[ascending chain]] of left ideals''' in a ring ''R''; i.e., <math>S</math> is a totally ordered set and <math>\mathfrak{a}_i \subset \mathfrak{a}_j</math> for each {{tmath|1= i < j }}. Then the union <math>\textstyle \bigcup_{i \in S} \mathfrak{a}_i</math> 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 <math>E \subset R</math> is a possibly empty subset and <math>\mathfrak{a}_0 \subset R</math> is a left ideal that is disjoint from ''E'', then there is an ideal that is maximal among the ideals containing <math>\mathfrak{a}_0</math> and disjoint from ''E''. (Again this is still valid if the ring ''R'' lacks the unity 1.) When <math>R \ne 0</math>, taking <math>\mathfrak{a}_0 = (0)</math> and {{tmath|1= 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. 
*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 {{tmath|1= RX }}. Such an ideal exists since it is the intersection of all left ideals containing ''X''. Equivalently, <math>RX</math> is the set of all the [[linear combinations|(finite) left ''R''-linear combinations]] of elements of ''X'' over ''R'': {{br}}<math display="block">RX = \{r_1x_1+\dots+r_nx_n \mid n\in\mathbb{N}, r_i\in R, x_i\in X\}</math>(since such a span is the smallest left ideal containing ''X''.)<ref group = note>If ''R'' does not have a unit, then the internal descriptions above must be modified slightly. In addition to the finite sums of products of things in ''X'' with things in ''R'', we must allow the addition of ''n''-fold sums of the form {{nowrap|''x'' + ''x'' + ... + ''x''}}, and ''n''-fold sums of the form {{nowrap|(−''x'') + (−''x'') + ... + (−''x'')}} for every ''x'' in ''X'' and every ''n'' in the natural numbers. When ''R'' has a unit, this extra requirement becomes superfluous.</ref> 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., {{br}}<math display="block">RXR = \{r_1x_1s_1+\dots+r_nx_ns_n \mid n\in\mathbb{N}, r_i\in R,s_i\in R, x_i\in X\} .</math>
* 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 <math>Rx</math> (resp. {{tmath|1= xR, RxR }}). The principal two-sided ideal <math>RxR</math> is often also denoted by {{tmath|1= (x) }}. If <math>X = \{ x_1, \dots, x_n \}</math> is a finite set, then <math>RXR</math> is also written as {{tmath|1= (x_1, \dots, x_n) }}.
* There is a bijective correspondence between ideals and [[congruence relation]]s (equivalence relations that respect the ring structure) on the ring: Given an ideal <math>I</math> of a ring {{tmath|1= R }}, let <math>x\sim y</math> if {{tmath|1= x-y\in I }}.  Then <math>\sim</math> is a congruence relation on {{tmath|1= R }}. Conversely, given a congruence relation <math>\sim</math> on {{tmath|1= R }}, let {{tmath|1= I=\{x\in R:x\sim 0\} }}.  Then <math>I</math> is an ideal of {{tmath|1= R }}.