practice to consider only localizations by multiplicative sets. For example, the localization by a single element introduces fractions of the form but also products of such fractions, such as So, the denominators will belong to the multiplicative set of the powers of . Therefore, one generally talks of "the localization by the powers of an element" rather than of "the localization by an element". The localization of a ring by a multiplicative set is generally denoted but other notations are commonly used in some special cases: if consists of the powers of a single element, is often denoted if is the complement of a prime ideal , then is denoted In the remainder of this article, only localizations by a multiplicative set are considered. Integral domains When the ring is an integral domain and does not contain , the ring is a subring of the field of fractions of . As such, the localization of a domain is a domain. More precisely, it is the subring of the field of fractions of , that consists of the fractions such that This is a subring since the sum and the product of two elements of are in This results from the defining property of a multiplicative set, which implies also that In this case, is a subring of It is shown below that this is no longer true in general, typically when contains zero divisors. For example, the decimal fractions are the localization of the ring of integers by the multiplicative set of the powers of ten. In this case, consists of the rational numbers that can be written as where is an integer, and is a nonnegative integer. General construction In the general case, a problem arises with zero divisors. Let be a multiplicative set in a commutative ring . Suppose that and is a zero divisor with Then is the image in of and one has Thus some nonzero elements of must be zero in The construction that follows is designed for taking this into account. Given and as above, one considers the equivalence relation on that is defined by if there exists a such that The localization is defined as the set of the equivalence classes for this relation. The class of is denoted as or So, one has if and only if there is a such that The reason for the is to handle cases such as