Ideal (algebra)

The ideal is one of the basic concepts of general algebra . Ideals have the greatest value in ring theory , but are also defined for semigroups , algebras, and some other algebraic structures . The name “ideal” is derived from the “ ideal numbers ”, which were introduced in 1847 by the German mathematician EE Kummer ^[1] . The simplest example of an ideal is a subring of even numbers in the ring of integers . Ideals provide a convenient language for generalizing the results of number theory to common rings.

For example, instead of simple numbers , simple ideals are studied in rings , mutually simple ideals are introduced as a generalization of mutually simple numbers , it is possible to prove an analogue of the Chinese remainder theorem for ideals.

In some important class of rings (the so-called Dedekind ones ) one can even get an analogue of the main theorem of arithmetic : in these rings each non-zero ideal can be uniquely represented as a product of simple ideals.

Content

Definition

For ring $R$ ${\ displaystyle R}$ an ideal is called a subring , closed relative to multiplication by elements from $R$ ${\ displaystyle R}$ . Moreover, an ideal is called left (respectively, right ) if it is closed with respect to multiplication from the left (respectively, to the right) to elements from $R$ ${\ displaystyle R}$ . The ideal, which is both left and right, is called two-sided . The two-sided ideal is often called just the ideal . In the commutative case, all these three concepts coincide and the term ideal is always used.

More precisely: The ideal ring $R$ ${\ displaystyle R}$ called such a subring $I$ ${\ displaystyle I}$ rings $R$ ${\ displaystyle R}$ , what

$\forall i\in I\;\forall r\in R$ ${\ displaystyle \ forall i \ in I \; \ forall r \ in R}$ composition $ir\in I$ ${\ displaystyle ir \ in I}$ (condition on right ideals);
$\forall i\in I\;\forall r\in R$ ${\ displaystyle \ forall i \ in I \; \ forall r \ in R}$ composition $ri\in I$ ${\ displaystyle ri \ in I}$ (condition on left ideals).

Similarly, for a semigroup, its ideal is a subsemigroup, for which any of these conditions is true (or both for a two-sided ideal), the same for algebra.

Note

For $R$ ${\ displaystyle R}$ -algebras $A$ ${\ displaystyle A}$ ( algebra over the ring $R$ ${\ displaystyle R}$ a) ideal ring $A$ ${\ displaystyle A}$ may, generally speaking, not be an ideal of algebra $A$ ${\ displaystyle A}$ , since this subring is not necessarily a subalgebra , that is, also a submodule over $R$ ${\ displaystyle R}$ . For example, if $A$ ${\ displaystyle A}$ there is $k$ ${\ displaystyle k}$ -algebra with zero multiplication, then the set of all ideals of a ring $A$ ${\ displaystyle A}$ coincides with the set of all subgroups of an additive group $A$ ${\ displaystyle A}$ and the set of all ideals of algebra $A$ ${\ displaystyle A}$ coincides with the set of all subspaces of the vector $k$ ${\ displaystyle k}$ -spaces $A$ ${\ displaystyle A}$ . However, in the case where $A$ ${\ displaystyle A}$ - algebra with unit, both of these concepts are the same.

Related definitions

For any ring $R$ ${\ displaystyle R}$ self $R$ ${\ displaystyle R}$ and zero ideal $0$ ${\ displaystyle 0}$ are ideals (bilateral). Such ideals are called trivial . Own ideals are ideals that form their own subset , that is, they do not coincide with everything. $R$ ${\ displaystyle R}$ ^[2] ^[3] .
Many classes of rings and algebras are determined by the conditions on their ideal or lattice of ideals. For example:
- A ring that does not have non-trivial two-sided ideals is called simple .
- A ring that does not have non-trivial ideals (not necessarily bilateral) is a body . See also: ring of main ideals , Artinian ring , Noetherian ring .

The topological space is connected with any commutative ring with unity . $\operatorname {Spec} A$ ${\ displaystyle \ operatorname {Spec} A}$ - spectrum of a ring whose points are all simple ideals of a ring $A$ ${\ displaystyle A}$ other than $A$ ${\ displaystyle A}$ , and closed sets are defined as sets of simple ideals containing some set $E$ ${\ displaystyle E}$ ring elements $A$ ${\ displaystyle A}$ (or, equivalently, ideal $I$ ${\ displaystyle I}$ generated by this set). This topology is called the Zariski topology .

The concept of an ideal is closely related to the concept of a module . An ideal (right or left) can be defined as a submodule of a ring, considered as a right or left module over itself.

Properties

Left ideals in R are right ideals in the so-called. opposite ring $R^{0}$ ${\ displaystyle R ^ {0}}$ - a ring with the same elements and the same addition as this one, but with a certain multiplication $a*b=ba$ ${\ displaystyle a * b = ba}$ , and vice versa.
Bilateral ideals in rings and algebras play the same role as normal subgroups in groups :
- For every homomorphism $f:A\to B$ ${\ displaystyle f: A \ to B}$ the core $\operatorname {Ker} f$ ${\ displaystyle \ operatorname {Ker} f}$ is an ideal, and vice versa, every ideal is the kernel of some homomorphism.
- Moreover, the ideal uniquely (up to isomorphism ) determines the image of the homomorphism, the kernel of which it is: $f(A)$ ${\ displaystyle f (A)}$ isomorphic to a factor ring ( factorialgebra ) $A/I$ ${\ displaystyle A / I}$ .
In the ring $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ integers all ideals are prime and have the form $n\mathbb {Z} =\{nz|z\in \mathbb {Z} \}$ ${\ displaystyle n \ mathbb {Z} = \ {nz | z \ in \ mathbb {Z} \}}$ where $n\in \mathbb {N} _{0}$ ${\ displaystyle n \ in \ mathbb {N} _ {0}}$ .
The intersection of ideals is also an ideal (often, especially in commutative algebra, the intersection is called the smallest common multiple ).

Types of ideals

Main ideal : Ideal generated by one element.
The ultimate ideal
Minimal ideal
Maximum ideal : A proper ideal I is said to be maximal if there is no proper ideal J such that I is a proper subset of J. The quotient ring on the maximal ideal is a field .
Modular ideal
Nilpotent ideal
Primary ideal
Primary ideal
Simple ideal
Radical ideal : An ideal that coincides with its radical .

Basic Structures

The main ideals . If p belongs to R , and k is any integer, then $\{pr+kp:\,r\in R\ ,k\in \mathbb {Z} \}$ ${\ displaystyle \ {pr + kp: \, r \ in R \, k \ in \ mathbb {Z} \}}$ - will be a minimal right ideal containing p , and $\{rp+kp:\,r\in R\ ,k\in \mathbb {Z} \}$ ${\ displaystyle \ {rp + kp: \, r \ in R \, k \ in \ mathbb {Z} \}}$ - minimal left ideal in R. They are called, respectively, the principal right and left ideals generated by p . In the commutative case, these ideals coincide and are also denoted by (p) . If the ring R contains a single element, then $kp=(k*1)p=p(k*1)$ ${\ displaystyle kp = (k * 1) p = p (k * 1)}$ , the main ideals generated by a can be written $pR=\{pr:\,r\in R\}$ ${\ displaystyle pR = \ {pr: \, r \ in R \}}$ and $Rp=\{rp:\,r\in R\}$ ${\ displaystyle Rp = \ {rp: \, r \ in R \}}$ respectively. Every ideal containing an element p contains the main ideal generated by it.

Ideal generated by many elements. The intersection of an arbitrary family of left ideals of the ring R is a left ideal of the ring R. Therefore, for every subset M of the ring R, there exists a minimal left ideal containing it, namely, the intersection of all left ideals containing the set M. (The same is true for right and two-sided ideals.) For a ring R with a unit element, the minimal left ideal is a set of finite sums of the form $r_{1}m_{1}+\ldots +r_{n}m_{n}$ ${\ displaystyle r_ {1} m_ {1} + \ ldots + r_ {n} m_ {n}}$ , the minimal right ideal is the set of finite sums of the form $m_{1}r_{1}+\ldots +m_{n}r_{n}$ ${\ displaystyle m_ {1} r_ {1} + \ ldots + m_ {n} r_ {n}}$ , minimal two-sided ideal - a set of finite sums of the form $r_{1}m_{1}r'_{1}+\ldots +r_{n}m_{n}r'_{n}$ ${\ displaystyle r_ {1} m_ {1} r '_ {1} + \ ldots + r_ {n} m_ {n} r' _ {n}}$ , where m _i are arbitrary elements of the set M , and r _i , r ' _i are arbitrary elements of the ring R. If the ring does not contain ones, then the minimal left ideal will be $r_{1}m_{1}+\ldots +r_{n}m_{n}+k_{1}m'_{1}+\ldots +k_{s}m'_{s}$ ${\ displaystyle r_ {1} m_ {1} + \ ldots + r_ {n} m_ {n} + k_ {1} m '_ {1} + \ ldots + k_ {s} m' _ {s}}$ minimum right $m_{1}r_{1}+\ldots +m_{n}r_{n}+k_{1}m'_{1}+k_{2}m'_{2}+\ldots +k_{s}m'_{s}$ ${\ displaystyle m_ {1} r_ {1} + \ ldots + m_ {n} r_ {n} + k_ {1} m '_ {1} + k_ {2} m' _ {2} + \ ldots + k_ {s} m '_ {s}}$ minimum bilateral $r_{1}m_{1}r'_{1}+\ldots +r_{n}m_{n}r'_{n}+k_{1}r''_{1}m'_{1}+\ldots +k_{s}r''_{s}m'_{s}+k'_{1}m''_{1}r'''_{1}+\ldots +k'_{t}m''_{t}r'''_{t}+k''_{1}m'''_{1}+\ldots +k''_{w}m''''_{w}$ ${\ displaystyle r_ {1} m_ {1} r '_ {1} + \ ldots + r_ {n} m_ {n} r' _ {n} + k_ {1} r '' _ {1} m'_ {1} + \ ldots + k_ {s} r '' _ {s} m '_ {s} + k' _ {1} m '' _ {1} r '' '_ {1} + \ ldots + k '_ {t} m' '_ {t} r' '' _ {t} + k '' _ {1} m '' '_ {1} + \ ldots + k' '_ {w} m' '' '_ {w}}$ where is everyone $k_{i}(k'_{i})$ ${\ displaystyle k_ {i} (k '_ {i})}$ - any integers. These ideals are called generated by the set M. In the commutative case, they all coincide and are denoted as: (M) . Ideals generated by a finite set are called finitely generated .

The sum of ideals. If an arbitrary family of ideals is given in the ring R $I_{\alpha }$ ${\ displaystyle I _ {\ alpha}}$ their sum $\sum I_{\alpha }$ ${\ displaystyle \ sum I _ {\ alpha}}$ called the minimal ideal that contains them all. It is generated by the union of these ideals, and its elements are any finite sums of elements from their union (the union of ideals itself is usually not an ideal). Regarding the sum, all (left, right, or two-sided) ideals of a ring (or algebra) form a lattice . Each ideal is the sum of the main ideals. Often, especially in commutative algebra, the sum is called the greatest common divisor).

The intersection of ideals (as the intersection of sets ) is always an ideal. On the other hand, the union of two ideals is an ideal only when one of them is a subset of the other. Indeed let ${\mathfrak {a}}$ ${\ displaystyle {\ mathfrak {a}}}$ and ${\mathfrak {b}}$ ${\ displaystyle {\ mathfrak {b}}}$ - two (left) ideals, neither of which is a subset of the other, and ${\mathfrak {a}}\cup {\mathfrak {b}}$ ${\ displaystyle {\ mathfrak {a}} \ cup {\ mathfrak {b}}}$ is a left ideal. In this case, obviously ${\mathfrak {a}}\cup {\mathfrak {b}}$ ${\ displaystyle {\ mathfrak {a}} \ cup {\ mathfrak {b}}}$ - the smallest ideal containing ${\mathfrak {a}}$ ${\ displaystyle {\ mathfrak {a}}}$ and ${\mathfrak {b}}$ ${\ displaystyle {\ mathfrak {b}}}$ , i.e ${\mathfrak {a}}\cup {\mathfrak {b}}={\mathfrak {a}}+{\mathfrak {b}}$ ${\ displaystyle {\ mathfrak {a}} \ cup {\ mathfrak {b}} = {\ mathfrak {a}} + {\ mathfrak {b}}}$ . There is an element $a\in {\mathfrak {a}},a\notin {\mathfrak {b}}$ ${\ displaystyle a \ in {\ mathfrak {a}}, a \ notin {\ mathfrak {b}}}$ . Then for any $b\in {\mathfrak {b}}\;a+b\notin {\mathfrak {b}}$ ${\ displaystyle b \ in {\ mathfrak {b}} \; a + b \ notin {\ mathfrak {b}}}$ as in this case $a\in {\mathfrak {b}}$ ${\ displaystyle a \ in {\ mathfrak {b}}}$ , Consequently, $a+b\in {\mathfrak {a}}$ ${\ displaystyle a + b \ in {\ mathfrak {a}}}$ and $b\in {\mathfrak {a}}$ ${\ displaystyle b \ in {\ mathfrak {a}}}$ , so ${\mathfrak {b}}\subset {\mathfrak {a}}$ ${\ displaystyle {\ mathfrak {b}} \ subset {\ mathfrak {a}}}$ - a contradiction.

The product of ideals. The product of ideals I and J is the ideal IJ , generated by all products ab , where a is an element of the ideal I , b is an element of the ideal J. The infinite product of ideals is not defined.

Private ideals. In the commutative ring for the ideal I , non-zero, and the ideal J , their quotient is defined — the ideal $I^{-1}J=\{x\in R\colon \,\forall i\in I\,ix\in J\}$ ${\ displaystyle I ^ {- 1} J = \ {x \ in R \ colon \, \ forall i \ in I \, ix \ in J \}}$ . This ideal is called the annihilator of the ideal I in the case when J = (0) ,.

The radical of the ideal I is the set ${\sqrt {I}}=\{f\in A:\,\exists n\in \mathbb {N} \,\,f^{n}\in {I}\}$ ${\ displaystyle {\ sqrt {I}} = \ {f \ in A: \, \ exists n \ in \ mathbb {N} \, \, f ^ {n} \ in {I} \}}$ . It is also an ideal of the ring A , if only the ring A is commutative. In the case when I = (0) , this ideal is called the nilradical of the ring A. Its elements are all nilpotent ring elements. If a commutative ring has no nilpotent elements, except zero (it has a zero nilradical), then it is called radical . An ideal I is called radical if it coincides with its radical. In this case, the factor ring R / I has no nilpotent elements except zero.

Inductive limit . If a family (chain) of ideals is given $\{I_{\alpha }\}_{\alpha \in A}$ ${\ displaystyle \ {I _ {\ alpha} \} _ {\ alpha \ in A}}$ , numbered by a linearly ordered set A , so that for any indices $\alpha <\beta$ ${\ displaystyle \ alpha <\ beta}$ of A ideal $I_{\alpha }$ ${\ displaystyle I _ {\ alpha}}$ ideally contained $I_{\beta }$ ${\ displaystyle I _ {\ beta}}$ , then their union is an ideal - the inductive limit of a given chain of ideals. This ideal also coincides with the sum of all ideals from the chain. The fact that the inductive limit always exists means that the set of all ideals of the ring R is inductively ordered, and the Zorn lemma applies to it. It is often used to construct maximal ideals with some additional properties (see maximal ideal , simple ideal , ring of principal ideals ).

The image of an ideal under a homomorphism. Usually, the image of an ideal under a homomorphism is NOT an ideal, but if the homomorphism is surjective, then it is. In particular, since the factorization homomorphism is always surjective, with factorization each ideal becomes an ideal.

The prototype of an ideal under a homomorphism . If a $f:\,A\to B$ {\ displaystyle f: \, A \ to B} - ring homomorphism , its kernel $\operatorname {Ker} f=\{a\in A:\,f(a)=0\}$ ${\ displaystyle \ operatorname {Ker} f = \ {a \ in A: \, f (a) = 0 \}}$ is a two-sided ideal. More generally, if I is an arbitrary ideal in the ring B , its full preimage $f^{-1}I=\{a\in A:\,f(a)\in I\}$ ${\ displaystyle f ^ {- 1} I = \ {a \ in A: \, f (a) \ in I \}}$ is an ideal (left, right or two-sided, depending on what ideal I is ).

Homomorphism of factorization by an ideal. If I is a two-sided ideal in the ring R , then it is possible to determine the equivalence relation on R by the rule: x ~ y if and only if the difference xy belongs to I. It is verified that if, in a sum or product, one of the operands is replaced by an equivalent, the new result will be equivalent to the original one. Thus, the operations of addition and multiplication become defined on the set of R / I equivalence classes, turning it into a ring (commutativity and the presence of a unit are transferred from the ring R , if they exist). Simultaneously with this ring, a factorization homomorphism (canonical homomorphism) is defined. $\pi :\,R\to R/I$ ${\ displaystyle \ pi: \, R \ to R / I}$ which each element a of R associates with an equivalence class in which it is contained. The equivalence class of a is a set of elements of the form a + i for all i of the ideal I , therefore it is denoted by a + I , but sometimes the general notation for the equivalence class [a] is used . therefore $\pi (a)=[a]=a+I$ ${\ displaystyle \ pi (a) = [a] = a + I}$ . In this case, the ring R / I is called the factor ring of the ring R according to the ideal I.

History

Ideals were first introduced by Dedekind in 1876 in the third edition of his book Lectures on Number Theory. This was a generalization of the concept of ideal numbers introduced by Kummer .

In the future, these ideas were developed by Hilbert and especially Noether .

Links

Vinberg, E. B. Algebra Course, - M .: Factorial Press, 2002, ISBN 5-88688-060-7 .
Zarissky O., Samuel P. Commutative Algebra, T. 1—2, - M .: IL, 1963.
Leng S. Algebra, - M .: Mir, 1968.

Notes

↑ Ideal // Kazakhstan. National Encyclopedia . - Almaty: Kazakh Encyclopedias , 2005. - T. II. - ISBN 9965-9746-3-2 .
↑ ' Margherita Barile . Proper Ideal (eng.) On the Wolfram MathWorld website.
↑ Lecture on algebra at the Moscow State University

[1] Ideal // Kazakhstan. National Encyclopedia . - Almaty: Kazakh Encyclopedias , 2005. - T. II. - ISBN 9965-9746-3-2 .

[2] ' Margherita Barile . Proper Ideal (eng.) On the Wolfram MathWorld website.

[3] Lecture on algebra at the Moscow State University