to some elements. This is generally impossible for non-commutative monoids, but, in a commutative monoid, it is possible to add inverses to the elements that have the cancellation property (an element has the cancellation property if implies and implies This extension of a monoid is allowed by Grothendieck group construction. This is the method that is commonly used for constructing integers from natural numbers, rational numbers from integers and, more generally, the field of fractions of an integral domain, and localizations of commutative rings. In rings A ring is an algebraic structure with two operations, addition and multiplication, which are denoted as the usual operations on numbers. Under addition, a ring is an abelian group, which means that addition is commutative and associative; it has an identity, called the additive identity, and denoted ; and every element has an inverse, called its additive inverse and denoted . Because of commutativity, the concepts of left and right inverses are meaningless since they do not differ from inverses. Under multiplication, a ring is a monoid; this means that multiplication is associative and has an identity called the multiplicative identity and denoted . An invertible element for multiplication is called a unit. The inverse or multiplicative inverse (for avoiding confusion with additive inverses) of a unit is denoted or, when the multiplication is commutative, The additive identity is never a unit, except when the ring is the zero ring, which has as its unique element. If is the only non-unit, the ring is a field if the multiplication is commutative, or a division ring otherwise. In a noncommutative ring (that is, a ring whose multiplication is not commutative), a non-invertible element may have one or several left or right inverses. This is, for example, the case of the functions from the integers to themselves, which form a ring for pointwise operations; see above, . A commutative ring (that is, a ring whose multiplication is commutative) may be extended by adding inverses to elements that are not zero divisors (that is, their product with a nonzero element cannot be ). This is the process of localization, which produces, in particular, the field of rational numbers from the ring of integers, and, more generally, the field of fractions of an integral domain. Localization is also used with zero divisors, but, in this case the original ring is not a subring of the localisation; instead, it