Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept generalizes in an appropriate way some important properties of integers like "even number" or "multiple of 3".
For instance, in rings one studies prime ideals instead of prime numbers, one defines coprime ideals as a generalization of coprime numbers, and one can prove a generalized Chinese remainder theorem about ideals. In a certain class of rings important in number theory, the Dedekind domains, one can even recover a version of the fundamental theorem of arithmetic: in these rings, every nonzero ideal can be uniquely written as a product of prime ideals.
An ideal can be used to construct a factor ring in a similar way as a normal subgroup in group theory can be used to construct a factor group. The concept of an order ideal in order theory is derived from the notion of ideal in ring theory.
Contents
History
Ideals were first proposed by Dedekind in 1876 in the third edition of his book Vorlesungen über Zahlentheorie (Engl.: Lectures on number theory). They were a generalization of the concept of ideal numbers developed by Ernst Kummer. Later the concept was expanded by David Hilbert and especially Emmy Noether.
Definitions
Let R be a ring and with (R,+) the abelian group of the ring. Then a subset I of R is called right ideal if
- (I,+) is a subgroup of (R,+)
- for all i in I and all r in R : i r is still in I
and left ideal if
- (I,+) is a subgroup of (R,+)
- for all i in I and all r in R : r i is still in I
When R is a commutative ring the notion of left ideal and right ideal coincide and the two-sided ideal is simply called ideal. To keep the following definitions shorter we will only consider commutative rings.
We call I a proper ideal if it is a proper subset of R, that is, I does not equal R. A proper ideal I is called a maximal ideal if there exists no other proper ideal J with I a subset of J. A proper ideal I is called a prime ideal if for all ab in I, either a or b is in I.
If we can write every element x of I as
where i_{k} is an element of I and {r_{k}: k=1,...,n} is a fixed finite subset of R we say I is finitely generated. If I is generated by only one element we call I a principal ideal.
If A is any subset of the ring R, then we can define the ideal generated by A to be the smallest ideal of R containing A; it is denoted by <A> or (A) and contains all finite sums of the form
- r_{1}a_{1}s_{1} + ··· + r_{n}a_{n}s_{n}
with each r_{i} and s_{i} in R and each a_{i} in A. The principal ideals mentioned above are the special case when A is just the singleton {a}.
Examples
- The even integers form an ideal in the ring Z of all integers; it is usually denoted by 2Z. This because the sum of any even integers is even, and the product of any integer with an even integer is also even.
- In the ring Z of integers, every ideal can be generated by a single number (so Z is a principal ideal domain), and the ideal determines the number up to its sign.The concepts of "ideal" and "number" are therefore almost identical in Z (and in any principal ideal domain).
- The set of all polynomials with real coefficients which are divisible by the polynomial x^{2} + 1 is an ideal in the ring of all polynomials.
- The set of all n-by-n matrices whose last column is zero forms a left ideal in the ring of all n-by-n matrices. It is not a right ideal. The set of all n-by-n matrices whose last row is zero forms a right ideal but not a left ideal.
- The ring C(R) of all continuous functions f from R to R contains the ideal of all continuous functions f such that f(1) = 0. Another ideal in C(R) 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 |x| > L.
- {0} and R are ideals in every ring R. If R is commutative, then R is a field iff it has precisely two ideals, {0} and R.
Notes
An ideal is proper iff it does not contain 1.
The ideals can be partially ordered via subset inclusion and therefore as a consequence of Zorn's lemma every ideal is contained in a maximal ideal.
Every maximal ideal is prime. Maximal ideals can be directly characterized to be those ideals which are subsets of only two ideals: the improper ideal and the maximal ideal itself.
Because zero belongs to it, any ideal is nonempty.
The ring R can be considered as a left module over itself, and the left ideals of R are then seen as the submodules of this module. Similarly, the right ideals are submodules of R as a right module over itself, and the two-sided ideals are submodules of R as a bimodule over itself. If R is commutative, then all three sorts of module are the same, just as all three sorts of ideal are the same.
Factor rings (quotient rings) and kernels
Ideals are important because they appear as the kernels of ring homomorphisms and allow one to define factor rings, as will be described next.
Recall that a function f from R to S is a ring homomorphism iff f(a + b) = f(a) + f(b), f(ab) = f(a) f(b) for all a, b in R, and f(1) = 1. Then the kernel of f is defined as
- ker(f) := {a in R : f(a) = 0}.
The kernel is always a two-sided ideal of R.
Conversely, if we start with a two-sided ideal I of R, then we may define a congruence relation ~ on R as follows: a ~ b if and only if b − a is in I. In case a ~ b, we say that a and b are congruent modulo I. The equivalence class of the element a in R is given by
- [a] = a + I := { a + r : r in I }.
The set of all such equivalence classes is denoted by R/I; it becomes a ring, the factor ring or quotient ring of R modulo I, if one defines
- (a + I) + (b + I) = (a + b) + I;
- (a + I) * (b + I) = (ab) + I.
(Compare coset and quotient group.)
(Note that these quotient rings are unrelated to the quotient field, or field of fractions, of an integral domain, and also unrelated to the rings of quotients resulting from localization of rings.)
The map p from R to R/I defined by p(a) = a + I is a surjective ring homomorphism (or ring epimorphism) whose kernel is the original ideal I. In summary, we see that ideals are precisely the kernels of ring homomorphisms.
If R is commutative and I is a maximal ideal, then the factor ring R/I is a field; if I is only a prime ideal, then R/I is only an integral domain.
The most extreme examples of factor rings are provided by modding out by the most extreme ideals, {0} and R itself. R/{0} is naturally isomorphic to R, and R/R is the trivial ring {0}.
Ideal operations
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 lattice. Also, the union of two ideals is a subset of the sum of those two ideal. The reason for this is: for any element a inside an ideal, we can write it as a+0, or 0+a, therefore, it is contained in the sum as well.
The product of two ideals I and J is defined to be the ideal IJ generated by all products of the form ab with a in I and b in J. It is contained in the intersection of I and J.
Mathematically, the sum and product of ideals are defined as follows. For I and J ideals of R,
and
Important properties of these ideal operations are recorded in the Noether isomorphism theorems.