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Examples and properties

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The statement of the theorem

(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)(1) since it is precisely the two-sided ideal generated (see below) by the unity ⁠ 1R1_{R} ⁠. Also, the set {0R}\{0_{R}\} consisting of only the additive identity 0_{R} forms a two-sided ideal called the zero ideal and is denoted by ⁠ (0)(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 a{\mathfrak {a}} is proper if and only if it does not contain a unit element, since if u∈au\in {\mathfrak {a}} is a unit element, then r=(ru−1)u∈ar=(ru^{-1})u\in {\mathfrak {a}} for every ⁠ r∈Rr\in R ⁠. Typically there are plenty of proper ideals. In fact, if R is a skew-field, then (0),(1)(0),(1) are its only ideals and conversely: that is, a nonzero ring R is a skew-field if (0),(1)(0),(1) are the only left (or right) ideals. (Proof: if xx is a nonzero element, then the principal left ideal RxRx (see below) is nonzero and thus Rx=(1)Rx=(1) ; i.e., yx=1yx=1 for some nonzero ⁠ yy ⁠. Likewise, zy=1zy=1 for some nonzero zz . Then z=z(yx)=(zy)x=xz=z(yx)=(zy)x=x .) - The even integers form an ideal in the ring Z\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 ⁠ 2Z2\mathbb {Z} ⁠. More generally, the set of all integers divisible by a fixed integer nn is an ideal denoted ⁠ nZn\mathbb {Z} ⁠. In fact, every non-zero ideal of the ring Z\mathbb {Z} is generated by its smallest positive element, as a consequence of Euclidean division, so Z\mathbb {Z} is a principal ideal domain. - The set of all polynomials with real coefficients that are divisible by the polynomial x2+1x^{2}+1 is an ideal in the ring of all real-coefficient polynomials ⁠ R[x]\mathbb {R} [x] ⁠. - Take a ring RR and positive integer ⁠ nn ⁠. For each ⁠ 1≤i≤n1\leq i\leq n ⁠, the set of all n×nn\times n matrices with entries in RR whose ii -th row is zero is a right ideal in the ring Mn(R)M_{n}(R) of all n×nn\times n matrices with entries in ⁠ RR ⁠. It is not a left ideal. Similarly, for each ⁠ 1≤j≤n1\leq j\leq n ⁠, the set of all n×nn\times n matrices whose jj -th column is zero is a left ideal but not a right ideal. - The ring C(R)C(\mathbb {R} ) of all continuous functions ff from R\mathbb {R} to R\mathbb {R} under pointwise multiplication contains the ideal of all continuous functions ff such that ⁠ f(1)=0f(1)=0 ⁠. Another ideal in C(R)C(\mathbb {R} ) is given by those functions that vanish for large enough arguments, i.e. those continuous functions ff for which there exists a number L>0L>0 such that f(x)=0f(x)=0 whenever ⁠ ∣x∣>L\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)(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→Sf:R\to S is a ring homomorphism, then the kernel ker⁡(f)=f−1(0S)\ker(f)=f^{-1}(0_{S}) is a two-sided ideal of ⁠ RR ⁠. By definition, ⁠ f(1R)=1Sf(1_{R})=1_{S} ⁠, and thus if SS is not the zero ring (so ⁠ 1S≠0S1_{S}\neq 0_{S} ⁠), then ker⁡(f)\ker(f) is a proper ideal. More generally, for each left ideal I of S, the pre-image f−1(I)f^{-1}(I) is a left ideal. If I is a left ideal of R, then f(I)f(I) is a left ideal of the subring f(R)f(R) of S: unless f is surjective, f(I)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→Sf:R\to S ⁠, there is a bijective order-preserving correspondence between the left (resp. right, two-sided) ideals of RR containing the kernel of ff and the left (resp. right, two-sided) ideals of SS : the correspondence is given by I↦f(I)I\mapsto f(I) and the pre-image ⁠ J↦f−1(J)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⊂MS\subset M a subset, then the annihilator Ann⁡R(S)={r∈R∣rs=0,s∈S}\operatorname {Ann} _{R}(S)=\{r\in R\mid rs=0,s\in S\} of S is a left ideal. Given ideals a,b{\mathfrak {a}},{\mathfrak {b}} of a commutative ring R, the R-annihilator of (b+a)/a({\mathfrak {b}}+{\mathfrak {a}})/{\mathfrak {a}} is an ideal of R called the ideal quotient of a{\mathfrak {a}} by b{\mathfrak {b}} and is denoted by ⁠ (a:b)({\mathfrak {a}}:{\mathfrak {b}}) ⁠; it is an instance of idealizer in commutative algebra. - Let ai,i∈S{\mathfrak {a}}_{i},i\in S be an ascending chain of left ideals in a ring R; i.e., SS is a totally ordered set and ai⊂aj{\mathfrak {a}}_{i}\subset {\mathfrak {a}}_{j} for each ⁠ i<ji<j ⁠. Then the union ⋃i∈Sai\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⊂RE\subset R is a possibly empty subset and a0⊂R{\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 a0{\mathfrak {a}}_{0} and disjoint from E. (Again this is still valid if the ring R lacks the unity 1.) When R≠0R\neq 0 , taking a0=(0){\mathfrak {a}}_{0}=(0) and ⁠ E={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. - 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 RxRx (resp. ⁠ xR,RxRxR,RxR ⁠). The principal two-sided ideal RxRRxR is often also denoted by ⁠ (x)(x) ⁠ or ⟨x⟩\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 ⁠ RXRX ⁠. Such an ideal exists since it is the intersection of all left ideals containing X. Equivalently, RXRX is the set of all the (finite) left R-linear combinations of elements of X over R: RX={r1x1+⋯+rnxn∣n∈N,ri∈R,xi∈X}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={r1x1s1+⋯+rnxnsn∣n∈N,ri∈R,si∈R,xi∈X}.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={x1,…,xn}X=\{x_{1},\dots ,x_{n}\} is a finite set, then RXRRXR is also written as ⁠ (x1,…,xn)(x_{1},\dots ,x_{n}) ⁠ or ⟨x1,...,xn⟩\langle x_{1},...,x_{n}\rangle . More generally, the two-sided ideal generated by a (finite or infinite) set of indexed ring elements X={xi}i∈IX=\{x_{i}\}_{i\in I} is denoted (X)=(xi)i∈I(X)=(x_{i})_{i\in I} or ⟨X⟩=⟨xi⟩i∈I\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 II of a ring ⁠ RR ⁠, let x∼yx\sim y if ⁠ x−y∈Ix-y\in I ⁠. Then ∼\sim is a congruence relation on ⁠ RR ⁠. Conversely, given a congruence relation ∼\sim on ⁠ RR ⁠, let ⁠ I={x∈R:x∼0}I=\{x\in R:x\sim 0\} ⁠. Then II is an ideal of ⁠ RR ⁠. 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.