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 . In the ring of integers, every ideal can be generated by a single number (so is a principal ideal domain), as a consequence of Euclidean division (or some other way). 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. Radical ideal or semiprime ideal: A proper ideal I is called radical or semiprime if for any a in R, if an is in I for some n, then a is in I. The factor ring of a radical ideal is a semiprime ring for general rings, and is a reduced ring 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