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.
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
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
An ideal I is called proper if I ≠ R; 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.
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.