Ring ideal

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In abstract algebra, an ideal is a subset I of a ring R which is closed under the ring operations in the following sense:

  • for any a, b in I, we have a + b in I
  • for any a in I and r in R, we have ra in I and ar in I.

If the ring is not commutative, these ideals are sometimes called two-sided to distinguish them from the left-sided (where only ra in I is required in the second condition) and the right-sided ideals. 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. In commutative rings, the three concepts coincide.

Examples

  • The even integers form an ideal in the ring Z of all integers.
  • 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-sided ideal in the ring of all n-by-n matrices.
  • The ring C(R) of all continuous functions f : R -> R contains the ideal of all continuous functions f with f(1) = 0.
  • {0} and R are ideals in every ring R.

Factor rings and kernels

Ideals are important because they appear as the kernels of ring homomorphisms and allow to define factor rings, as will be described next.

If f : R -> S is a ring homomorphism, i.e. a function with 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, we may define an equivalence 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 + R = {a + r : r in R}

The set of all equivalence classes is denoted by R/I; it turns into a ring, the factor ring of R by I, if one defines

  • (a + R) + (b + R) = (a + b) + R and
  • (a + R) * (b + R) = (ab) + R.

The map p : R -> R/I defined by p(a) = a + R is a surjective ring homomorphism with kernel equal to I.

An ideal I is called proper if IR; it 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. If R is commutative and I is a maximal ideal, then R/I is a field. The only ideals in a field are {0} and the field itself.

Ideal operations

The sum and the intersection of ideals is again an ideal; with these two operations, the set of all ideals of a given ring forms a lattice.

If A is any subset of the ring R, we can define the ideal generated by A to be the smallest ideal of R containing A; it is denoted by <A> and contains all finite sums of the form ∑ riaisi with ri and si in R and ai in 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.



See also: