Quotient Rings
Xiamen University
05/04/2015
Quotient Rings
Quotient Rings
Definition. Let I be an ideal of a ring R, then with addition
+, I is a normal subgroup of (R, +). Then R/I is the set of
cosets, i.e. R/I = {a + I|a ∈ R} a quotient group with
respect for +. We write a + I by a. Define
a · b = ab
⇒ R/I forms a ring, called the quotient ring.
Quotient Rings
Quotient Rings
Theorem. Let I be an ideal of a ring R.
(a)
π : R → R/I
a 7→ a = a + I
is a ring homomorphism, called the canonical homomorphism.
(b) ker π = I.
Quotient Rings
Quotient Rings
Proposition. Let f : R → R0 be a ring homomorphism with
kernel I, and J is an ideal of R that is contained in I. Then
(a) There is a unique homomorphism: f : R/J → R0 such
that f ◦ π = f , i.e. the following diagram commutes:
/ R/J .
π
R
f
R0
}
f
(b) Imf = {f (a)|a ∈ R} is a subring of R0 , and
R/ ker f ∼
= Imf.
Quotient Rings
Quotient Rings
Definition. Elements in Z[i] are called Gauss integers.
Quotient Rings
Definition. Elements in Z[i] are called Gauss integers.
Proposition. The ring Z[i]/(1 + 3i) is isomorphic to the ring
Z/10Z of integers module 10.
Quotient Rings
Quotient Rings
Proof. Define
ϕ:
Z → Z[i]/(1 + 3i)
n 7→ n + (1 + 3i)
For any a + bi ∈ Z[i], we have
a + bi + (1 + 3i) = a + 3b + (1 + 3i) = ϕ(a + 3b), so ϕ is
surjective.
Next, we show ker ϕ = 10Z. It is clear that
ϕ(10n) = 10n+(1+3i) = (1+3i)(1−3i)n+(1+3i) ⊂ (1+3i),
i.e. 10Z ∈ ker ϕ. If n ∈ ker ϕ, then n = (1 + 3i)(a + bi) for
some Gauss integer a + bi. Hence n = a − 3b + (b + 3a)i, then
a − 3b = n, b + 3a = 0. So n = 10a ∈ 10Z, then ker ϕ = 10Z.
Quotient Rings
Quotient Rings
Correspondence Theorem. Let J be an ideal of R,
R̄ = R/J, and π : R → R̄ be the canonical map.
Quotient Rings
Correspondence Theorem. Let J be an ideal of R,
R̄ = R/J, and π : R → R̄ be the canonical map.
(a) There is a bijective correspondence between the set of
ideals of R which contains J and the set of all ideals in R̄,
given by
¯ ← I¯
I → π(I), π −1 (I)
Quotient Rings
Correspondence Theorem. Let J be an ideal of R,
R̄ = R/J, and π : R → R̄ be the canonical map.
(a) There is a bijective correspondence between the set of
ideals of R which contains J and the set of all ideals in R̄,
given by
¯ ← I¯
I → π(I), π −1 (I)
(b) If I ⊂ R corresponds to I¯ ⊂ R̄, then R/I and R̄/I¯
are isomorphic rings.
Quotient Rings
Homework
1.P356.6.3. Classify rings of order 10.
2.P356.6.8. Let I and J be ideals of a ring R such that
I + J = R.
(a) Prove that IJ = I ∩ J.
(c) Prove that if IJ = 0, then R is isomorphic to the product
(R/I) × (R/J).
3.P355.4.3. Prove that Z[i]/(2 + i) ∼
= Z/5Z.
4. Find all the ideals of Z/10Z.
Quotient Rings