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Integral Domains and Fraction Fields
Xiamen University
05/11/2015
Integral Domains and Fraction Fields
Integral Domains and Fraction Fields
Definition. An integral domain is a ring R having no
non-zero zero divisors. In other words, if a, b ∈ R such
that ab = 0, then a = 0 or b = 0.
Integral Domains and Fraction Fields
Definition. An integral domain is a ring R having no
non-zero zero divisors. In other words, if a, b ∈ R such
that ab = 0, then a = 0 or b = 0.
Lemma. If R is an integral domain, then the cancellation
law holds, i.e. if ab = ac and a 6= 0, then b = c.
Integral Domains and Fraction Fields
Integral Domains and Fraction Fields
Proposition. If R is an integral domain, then the polynomial
ring R[x] is also an integral domain.
Integral Domains and Fraction Fields
Proposition. If R is an integral domain, then the polynomial
ring R[x] is also an integral domain.
Proposition. If R is an integral domain and | R |<∝, then R
is a field.
Integral Domains and Fraction Fields
Integral Domains and Fraction Fields
Theorem. Let R be an integral domain. Then there exists an
embedding of R into a field F , that is, there is an injective
ring homomorphism from the integral domain R to a field F .
Integral Domains and Fraction Fields
Theorem. Let R be an integral domain. Then there exists an
embedding of R into a field F , that is, there is an injective
ring homomorphism from the integral domain R to a field F .
Remark. The field F is often called the field of fraction of
the integral domain R.
Integral Domains and Fraction Fields
Integral Domains and Fraction Fields
Proof. For a, b ∈ R and b 6= 0, we define a symbol a/b, called
a fraction. Two fractions a1 /b1 , a2 /b2 where b1 , b2 6= 0 are
said related, denoted by a1 /b1 ≈ a2 /b2 iff a1 b2 = a2 b1 .
It can be seen that the relation ≈ is an equivalence relation
on the set R0 of all fractions of R. Let
F = R0 / ≈ = {a/b | a, b ∈ R, b 6= 0 and a1 /b1 = a2 /b2 iff
a1 b2 = a2 b1 }. We claim that F is a field, where the addition
and multiplication are given by
a1 /b1 · a2 /b2 = a1 a2 /b1 b2
a1 /b1 + a2 /b2 = (a1 b2 + a2 b1 )/b1 b2
Now we can define a map
ϕ:
R −→ F
a 7−→ a/1
Clearly, ϕ is an injective ring homomorphism.
Integral Domains and Fraction Fields
Integral Domains and Fraction Fields
Example. Z is an integral domain, the field of fraction of Z is
the field Q of rational numbers.
Integral Domains and Fraction Fields
Example. Z is an integral domain, the field of fraction of Z is
the field Q of rational numbers.
Example. Let R[x] be the ring of real polynomials, then the
field of fractions of R[x] is rational function field is:
F = {f (x)/g(x) | f (x), g(x) ∈ R[x] and g(x) are nonzero
polynomials}
where the operations are:
f1 (x)/g1 (x) · f2 (x)/g2 (x) = f1 (x)f2 (x)/g1 (x)g2 (x)
f1 (x)/g1 (x) + f2 (x)/g2 (x) = f1 (x)g2 (x) + f2 (x)g1 (x)/g1 (x)g2 (x).
Integral Domains and Fraction Fields
Integral Domains and Fraction Fields
Proposition.(Universal property of the field of fractions)
Let R be an integral domain with field of fractions F . If
ϕ : R −→ K is an injective ring homomorphism from R to a
field K. Then there is a unique extension of ϕ to a ring
homomorphism ψ : F −→ K such that ψ |R = ϕ.
Integral Domains and Fraction Fields
Integral Domains and Fraction Fields
Example Let R be an integral domain. m and n are positive
integers and (m, n) = 1. For a, b ∈ D, if am = bm and
an = bn , prove that a = b.
Integral Domains and Fraction Fields
Homework
Integral Domains and Fraction Fields
Homework
1. Is there an integer domain containing exactly 10 elements?
2.(P357.7.5) A subset S of an integral domain R which is
closed under multiplication and which dose not contain 0 is
called a multiplicative set. Given a multiplicative set S, we
define S-fractions to be elements of the form a/b, where
b ∈ S. Show that the equivalence classes of S-fractions form
a ring.
3. Prove that R[x]/(x2 + 1) ∼
= C.
Integral Domains and Fraction Fields