Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal of a ring R is a subset I of R which is closed under R-linear combinations, in a sense made precise below. The concept of an order ideal that is known in order theory is derived from this notion is discussed in its dedicated article.

Definitions

To accommodate non-commutative rings, we must distinguish three cases: left ideals, right ideals, and two-sided ideals.

A subset I of the ring R is a left ideal of R if

1: the zero element 0 of R belongs to I

2: for any a,b in I, we have a + b in I, and

3L: for any a in I and r in R, we have ra in I

A subset I of R is a right ideal of R if, in addition to properties 1 and 2 from above, it satisfies

3R: for any a in I and r in R, we have ar in I

A subset which is both a left and a right ideal (that is, it satisfies properties 1, 2, 3L, and 3R) is called a two-sided ideal of R. The term ideal alone is usually short for "two-sided ideal".

If the ring R is commutative, then all three sorts of ideals are the same. If the ring is noncommutative, however, then they may be different.

Examples

• The even integers form an ideal in the ring Z of all integers; it is usually denoted by 2Z.
• 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-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.

Further properties of ideals

Because zero belongs to it, any ideal is nonempty. In fact, property 1 in the definition can be replaced with simply the requirement that I be nonempty.

Any left, right or two-sided ideal is a subgroup of the additive group (R,+).

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.

Types of ideals

The first two examples above are principal ideals; the principal (left) ideal generated by an element a in R is Ra := {ra : r in R}. The principal right ideal aR is defined similarly; and these two principal ideals generated by a are identical (and hence a two-sided ideal) if the ring is commutative. In that case, it's common to denote the principal ideal by <a> or (a).

An ideal I is called proper if I is not equal to R. An ideal is proper if and only if it doesn't contain 1. A proper ideal is called maximal if the only proper ideal it is contained in is itself. Every ideal is contained in a maximal ideal, a consequence of Zorn's lemma. A proper ideal I is called prime if, whenever ab belongs to I, then so does a or b (or both). Every maximal ideal is prime.

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) and 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.

(But 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 regular 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.

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

r₁a₁s₁ + ··· + r_na_ns_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}.

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.

Important properties of these ideal operations are recorded in the Noether isomorphism theorems.

Ideals as "ideal numbers"

The term "ideal" comes from the notion of ideal number: ideals were seen as a generalization of the concept of number. 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). In other rings, it turned out that the concept of "ideal" allows one to generalize several properties of numbers. For instance, in general 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.

See also:

• Modular arithmetic
• Chinese remainder theorem
• Noether isomorphism theorem
• Boolean prime ideal theorem