Abstract algebra concept
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field of rational numbers. Intuitively, it consists of ratios between integral domain elements.
The field of fractions of an integral domain
is sometimes denoted by
or
, and the construction is sometimes also called the fraction field, field of quotients, or quotient field of
. All four are in common usage, but are not to be confused with the quotient of a ring by an ideal, which is a quite different concept. For a commutative ring that is not an integral domain, the analogous construction is called the localization or ring of quotients.
Definition
Given an integral domain
and letting
, we define an equivalence relation on
by letting
whenever
. We denote the equivalence class of
by
. This notion of equivalence is motivated by the rational numbers
, which have the same property with respect to the underlying ring
of integers.
Then the field of fractions is the set
with addition given by
![{\displaystyle {\frac {n}{d}}+{\frac {m}{b}}={\frac {nb+md}{db}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and multiplication given by
![{\displaystyle {\frac {n}{d}}\cdot {\frac {m}{b}}={\frac {nm}{db}}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
One may check that these operations are well-defined and that, for any integral domain
,
is indeed a field. In particular, for
, the multiplicative inverse of
is as expected:
.
The embedding of
in
maps each
in
to the fraction
for any nonzero
(the equivalence class is independent of the choice
). This is modeled on the identity
.
The field of fractions of
is characterized by the following universal property:
- if
is an injective ring homomorphism from
into a field
, then there exists a unique ring homomorphism
that extends
.
There is a categorical interpretation of this construction. Let
be the category of integral domains and injective ring maps. The functor from
to the category of fields that takes every integral domain to its fraction field and every homomorphism to the induced map on fields (which exists by the universal property) is the left adjoint of the inclusion functor from the category of fields to
. Thus the category of fields (which is a full subcategory) is a reflective subcategory of
.
A multiplicative identity is not required for the role of the integral domain; this construction can be applied to any nonzero commutative rng
with no nonzero zero divisors. The embedding is given by
for any nonzero
.[1]
Examples
- The field of fractions of the ring of integers is the field of rationals:
. - Let
be the ring of Gaussian integers. Then
, the field of Gaussian rationals. - The field of fractions of a field is canonically isomorphic to the field itself.
- Given a field
, the field of fractions of the polynomial ring in one indeterminate
(which is an integral domain), is called the field of rational functions, field of rational fractions, or field of rational expressions[2][3][4][5] and is denoted
. - The field of fractions of the convolution ring of half-line functions yields a space of operators, including the Dirac delta function, differential operator, and integral operator. This construction gives an alternate representation of the Laplace transform that does not depend explicitly on an integral transform.[6]
Generalizations
Localization
For any commutative ring
and any multiplicative set
in
, the localization
is the commutative ring consisting of fractions
![{\displaystyle {\frac {r}{s}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
with
and
, where now
is equivalent to
if and only if there exists
such that
.
Two special cases of this are notable:
Note that it is permitted for
to contain 0, but in that case
will be the trivial ring.
Semifield of fractions
The semifield of fractions of a commutative semiring with no zero divisors is the smallest semifield in which it can be embedded.
The elements of the semifield of fractions of the commutative semiring
are equivalence classes written as
![{\displaystyle {\frac {a}{b}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
with
and
in
.
See also
References
- ^ Hungerford, Thomas W. (1980). Algebra (Revised 3rd ed.). New York: Springer. pp. 142–144. ISBN 3540905189.
- ^ Vinberg, Ėrnest Borisovich (2003). A course in algebra. American Mathematical Society. p. 131. ISBN 978-0-8218-8394-5.
- ^ Foldes, Stephan (1994). Fundamental structures of algebra and discrete mathematics. Wiley. p. 128. ISBN 0-471-57180-6.
- ^ Grillet, Pierre Antoine (2007). "3.5 Rings: Polynomials in One Variable". Abstract algebra. Springer. p. 124. ISBN 978-0-387-71568-1.
- ^ Marecek, Lynn; Mathis, Andrea Honeycutt (6 May 2020). Intermediate Algebra 2e. OpenStax. §7.1.
- ^ Mikusiński, Jan (14 July 2014). Operational Calculus. Elsevier. ISBN 9781483278933.