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The related, but distinct, concept of an ideal in order theory is derived from the notion of ideal in ring theory. A fractional ideal is a generalization of an ideal, and the usual ideals are sometimes called integral ideals for clarity.
Ernst Kummer invented the concept of ideal numbers to serve as the "missing" factors in number rings in which unique factorization fails; here the word "ideal" is in the sense of existing in imagination only, in analogy with "ideal" objects in geometry such as points at infinity.
In 1876, Richard Dedekind replaced Kummer's undefined concept by concrete sets of numbers, sets that he called ideals, in the third edition of Dirichlet's book Vorlesungen über Zahlentheorie, to which Dedekind had added many supplements.
Later the notion was extended beyond number rings to the setting of polynomial rings and other commutative rings by David Hilbert and especially Emmy Noether.
Definitions and motivation
For an arbitrary ring , let be its additive group. A subset is called a left ideal of if it is an additive subgroup of that "absorbs multiplication from the left by elements of "; that is, is a left ideal if it satisfies the following two conditions:
A right ideal is defined with the condition replaced by . A two-sided ideal is a left ideal that is also a right ideal, and is sometimes simply called an ideal. In the language of modules, the definitions mean that a left (resp. right, two-sided) ideal of R is precisely a left (resp. right, bi-) R-submodule of R when R is viewed as an R-module. When R is a commutative ring, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.
To understand the concept of an ideal, consider how ideals arise in the construction of rings of "elements modulo". For concreteness, let us look at the ring Zn of integers modulo a given integer n ? Z (note that Z is a commutative ring). The key observation here is that we obtain Zn by taking the integer line Z and wrapping it around itself so that various integers get identified. In doing so, we must satisfy two requirements: 1) n must be identified with 0 since n is congruent to 0 modulo n, and 2) the resulting structure must again be a ring. The second requirement forces us to make additional identifications (i.e., it determines the precise way in which we must wrap Z around itself). The notion of an ideal arises when we ask the question:
What is the exact set of integers that we are forced to identify with 0?
The answer is, unsurprisingly, the set of all integers congruent to 0 modulo n. That is, we must wrap Z around itself infinitely many times so that the integers ..., , , , , ... will all align with 0. If we look at what properties this set must satisfy in order to ensure that Zn is a ring, then we arrive at the definition of an ideal. Indeed, one can directly verify that nZ is an ideal of Z.
Remark. Identifications with elements other than 0 also need to be made. For example, the elements in must be identified with 1, the elements in must be identified with 2, and so on. Those, however, are uniquely determined by nZ since Z is an additive group.
We can make a similar construction in any commutative ring R: start with an arbitrary , and then identify with 0 all elements of the ideal It turns out that the ideal xR is the smallest ideal that contains x, called the ideal generated by x. More generally, we can start with an arbitrary subset , and then identify with 0 all the elements in the ideal generated by S: the smallest ideal (S) such that . The ring that we obtain after the identification depends only on the ideal (S) and not on the set S that we started with. That is, if , then the resulting rings will be the same.
Therefore, an ideal I of a commutative ring R captures canonically the information needed to obtain the ring of elements of R modulo a given subset . The elements of I, by definition, are those that are congruent to zero, that is, identified with zero in the resulting ring. The resulting ring is called the quotient of R by I and is denoted R/I. Intuitively, the definition of an ideal postulates two natural conditions necessary for I to contain all elements designated as "zeros" by R/I:
I is an additive subgroup of R: the zero 0 of R is a "zero" , and if and are "zeros", then is a "zero" too.
Any multiplied by a "zero" is a "zero" .
It turns out that the above conditions are also sufficient for I to contain all the necessary "zeros": no other elements have to be designated as "zero" in order to form R/I. (In fact, no other elements should be designated as "zero" if we want to make the fewest identifications.)
Remark. If R is not necessarily commutative, the above construction still works using two-sided ideals.
Examples and properties
For the sake of succinctness, 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 since it is precisely the two-sided ideal generated (see below) by the unity . Also, the set consisting of only the additive identity 0R forms a two-sided ideal called the zero ideal and is denoted by .[note 1] 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 is proper if and only if it does not contain a unit element, since if is a unit element, then for every . Typically there are plenty of proper ideals. In fact, if R is a skew-field, then are its only ideals and conversely: that is, a nonzero ring R is a skew-field if are the only left (or right) ideals. (Proof: if is a nonzero element, then the principal left ideal (see below) is nonzero and thus ; i.e., for some nonzero . Likewise, for some nonzero . Then .)
The even integers form an ideal in the ring of all integers; it is usually denoted by . This is because the sum of any even integers is even, and the product of any integer with an even integer is also even. Similarly, the set of all integers divisible by a fixed integer n is an ideal denoted .
The set of all polynomials with real coefficients which are divisible by the polynomial x2 + 1 is an ideal in the ring of all polynomials.
The set of all n-by-nmatrices whose last row is zero forms a right ideal in the ring of all n-by-n matrices. It is not a left ideal. The set of all n-by-n matrices whose last column is zero forms a left ideal but not a right ideal.
The ring of all continuous functionsf from to under pointwise multiplication contains the ideal of all continuous functions f such that f(1) = 0. Another ideal in is given by those functions which 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 || > L.
A ring is called a simple ring if it is nonzero and has no two-sided ideals other than . 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 is a ring homomorphism, then the kernel is a two-sided ideal of . By definition, , and thus if is not the zero ring (so ), then is a proper ideal. More generally, for each left ideal I of S, the pre-image is a left ideal. If I is a left ideal of R, then is a left ideal of the subring of S: unless f is surjective, need not be an ideal of S; see also #Extension and contraction of an ideal below.
Ideal correspondence: Given a surjective ring homomorphism , there is a bijective order-preserving correspondence between the left (resp. right, two-sided) ideals of containing the kernel of and the left (resp. right, two-sided) ideals of : the correspondence is given by and the pre-image . 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).
(For those who know modules) If M is a left R-module and a subset, then the annihilator of S is a left ideal. Given ideals of a commutative ring R, the R-annihilator of is an ideal of R called the ideal quotient of by and is denoted by ; it is an instance of idealizer in commutative algebra.
Let be an ascending chain of left ideals in a ring R; i.e., is a totally ordered set and for each . Then the union 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 is a possibly empty subset and is a left ideal that is disjoint from E, then there is an ideal that is maximal among the ideals containing and disjoint from E. (Again this is still valid if the ring R lacks the unity 1.) When , taking and , 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 . Such an ideal exists since it is the intersection of all left ideals containing X. Equivalently, is the set of all the (finite) left R-linear combinations of elements of X over R:
(since such a span is the smallest left ideal containing X.)[note 2] 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.,
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 (resp. ). The principal two-sided ideal is often also denoted by . If is a finite set, then is also written as .
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 if . Then ~ is a congruence relation on R. Conversely, given a congruence relation ~ on R, let . Then I is an ideal of R.
Types of ideals
To simplify the description all rings are assumed to be commutative. The non-commutative case is discussed in detail in the respective articles.
Ideals are important because they appear as kernels of ring homomorphisms and allow one to define factor rings. Different types of ideals are studied because they can be used to construct different types of factor rings.
Maximal ideal: A proper ideal I is called a maximal ideal if there exists no other proper ideal J with I a proper subset of J. The factor ring of a maximal ideal is a simple ring in general and is a field for commutative rings.
Minimal ideal: A nonzero ideal is called minimal if it contains no other nonzero ideal.
Prime ideal: A proper ideal I is called a prime ideal if for any a and b in R, if ab is in I, then at least one of a and b is in I. The factor ring of a prime ideal is a prime ring in general and is an integral domain for commutative rings.
Primary ideal: An ideal I is called a primary ideal if for all a and b in R, if ab is in I, then at least one of a and bn is in I for some natural numbern. Every prime ideal is primary, but not conversely. A semiprime primary ideal is prime.
Two other important terms using "ideal" are not always ideals of their ring. See their respective articles for details:
Fractional ideal: This is usually defined when R is a commutative domain with quotient fieldK. Despite their names, fractional ideals are R submodules of K with a special property. If the fractional ideal is contained entirely in R, then it is truly an ideal of R.
Invertible ideal: Usually an invertible ideal A is defined as a fractional ideal for which there is another fractional ideal B such that AB=BA=R. Some authors may also apply "invertible ideal" to ordinary ring ideals A and B with AB=BA=R in rings other than domains.
The sum and product of ideals are defined as follows. For and , left (resp. right) ideals of a ring R, their sum is
which is a left (resp. right) ideal,
and, if are two-sided,
i.e. the product is the ideal generated by all products of the form ab with a in and b in .
Note is the smallest left (resp. right) ideal containing both and (or the union ), while the product is contained in the intersection of and .
The distributive law holds for two-sided ideals ,
If a product is replaced by an intersection, a partial distributive law holds:
where the equality holds if contains or .
Remark: The sum and the intersection of ideals is again an ideal; with these two operations as join and meet, the set of all ideals of a given ring forms a completemodular lattice. The lattice is not, in general, a distributive lattice. The three operations of intersection, sum (or join), and product make the set of ideals of a commutative ring into a quantale.
If are ideals of a commutative ring R, then in the following two cases (at least)
is generated by elements that form a regular sequence modulo .
(More generally, the difference between a product and an intersection of ideals is measured by the Tor functor: )
An integral domain is called a Dedekind domain if for each pair of ideals , there is an ideal such that . It can then be shown that every nonzero ideal of a Dedekind domain can be uniquely written as a product of maximal ideals, a generalization of the fundamental theorem of arithmetic.
Examples of ideal operations
In we have
since is the set of integers which are divisible by both and .
Let and let . Then,
In the first computation, we see the general pattern for taking the sum of two finitely generated ideals, it is the ideal generated by the union of their generators. In the last three we observe that products and intersections agree whenever the two ideals intersect in the zero ideal. These computations can be checked using Macaulay2.
Radical of a ring
Ideals appear naturally in the study of modules, especially in the form of a radical.
For simplicity, we work with commutative rings but, with some changes, the results are also true for non-commutative rings.
Indeed, if is a simple module and x is a nonzero element in M, then and , meaning is a maximal ideal. Conversely, if is a maximal ideal, then is the annihilator of the simple R-module . There is also another characterization (the proof is not hard):
For a not-necessarily-commutative ring, it is a general fact that is a unit element if and only if is (see the link) and so this last characterization shows that the radical can be defined both in terms of left and right primitive ideals.
The following simple but important fact (Nakayama's lemma) is built-in to the definition of a Jacobson radical: if M is a module such that , then M does not admit a maximal submodule, since if there is a maximal submodule , and so , a contradiction. Since a nonzero finitely generated module admits a maximal submodule, in particular, one has:
If R is an Artinian ring, then is nilpotent and . (Proof: first note the DCC implies for some n. If (DCC) is an ideal properly minimal over the latter, then . That is, , a contradiction.)
Extension and contraction of an ideal
Let A and B be two commutative rings, and let f : A -> B be a ring homomorphism. If is an ideal in A, then need not be an ideal in B (e.g. take f to be the inclusion of the ring of integers Z into the field of rationals Q). The extension of in B is defined to be the ideal in B generated by . Explicitly,
If is an ideal of B, then is always an ideal of A, called the contraction of to A.
Assuming f : A -> B is a ring homomorphism, is an ideal in A, is an ideal in B, then:
is prime in B is prime in A.
It is false, in general, that being prime (or maximal) in A implies that is prime (or maximal) in B. Many classic examples of this stem from algebraic number theory. For example, embedding. In , the element 2 factors as where (one can show) neither of are units in B. So is not prime in B (and therefore not maximal, as well). Indeed, shows that , , and therefore .
The following is sometimes useful: a prime ideal is a contraction of a prime ideal if and only if . (Proof: Assuming the latter, note intersects , a contradiction. Now, the prime ideals of correspond to those in B that are disjoint from . Hence, there is a prime ideal of B, disjoint from , such that is a maximal ideal containing . One then checks that lies over . The converse is obvious.)
Ideals can be generalised to any monoid object, where is the object where the monoid structure has been forgotten. A left ideal of is a subobject that "absorbs multiplication from the left by elements of "; that is, is a left ideal if it satisfies the following two conditions:
A right ideal is defined with the condition "" replaced by "'". A two-sided ideal is a left ideal that is also a right ideal, and is sometimes simply called an ideal. When is a commutative monoid object respectively, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.
An ideal can also be thought of as a specific type of R-module. If we consider as a left -module (by left multiplication), then a left ideal is really just a left sub-module of . In other words, is a left (right) ideal of if and only if it is a left (right) -module which is a subset of . is a two-sided ideal if it is a sub--bimodule of .
Example: If we let , an ideal of is an abelian group which is a subset of , i.e. for some . So these give all the ideals of .
^Some authors call the zero and unit ideals of a ring R the trivial ideals of R.
^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 , and n-fold sums of the form for every x in X and every n in the natural numbers. When R has a unit, this extra requirement becomes superfluous.
^ abJohn Stillwell (2010). Mathematics and its history. p. 439.
^Harold M. Edwards (1977). Fermat's last theorem. A genetic introduction to algebraic number theory. p. 76.
^Everest G., Ward T. (2005). An introduction to number theory. p. 83.