Download Ideals, congruence modulo ideal, factor rings

Ideals, congruence modulo ideal, factor rings
Sergei Silvestrov
Spring term 2011, Lecture 6
Contents of the lecture
+ Homomorphisms of rings
+ Ideals
+ Factor rings
– Typeset by FoilTEX –
Algebra course FMA190/FMA190F
2011, Spring term 2011, Sergei Silvestrov lectures
Congruence in F[x] and congruence classes
This short repetition from previous Lecture 5 is important example for material on ideals and
factor rings, which is the main topic of this Lecture 6
F is a field. p(x) ∈ F[x] is the ring of polynomials.
Definition 1. (Sec 5.1, p. 119)
f ≡ g(mod p) are congruent modulo p if and only if p(x)∣( f (x) − g(x))
Theorem 1. (Theorem 5.1, p. 120) Congruence ≡ is equivalence relation on F[x].
Definition 2. Congruence classes are defined as equivalence classes corresponding to the equivalence relation ≡
[ f ] = {g : g ≡ f (mod p)} = { f (x) + k(x)p(x) : k(x) ∈ F[x]}
Theorem 2. (Th. 5.3, p. 121)
f (x) ≡ g(x)(mod p(x))
⇔
[ f ] = [g]
OBS! [0] = {g : g ≡ 0(mod p)} = {k(x)p(x) : k(x) ∈ F[x]} is closed under multiplication by
any h(x) ∈∈ F[x] since
h(x)(k(x)p(x)) = (h(x)k(x))p(x)
In the general terminology we will study in this lecture,
[0] is an ideal in F[x] and [ f ] is an element of a factor rings of the ring F[x] by the ideal [0].
– Typeset by FoilTEX –
1
Algebra course FMA190/FMA190F
2011, Spring term 2011, Sergei Silvestrov lectures
Ring homomorphisms
This repetition from previous Lectures is important for material on ideals and factor rings,
which is the main topic of this Lecture 6
Definition 3. If R and R′ are rings, a ring homomorphism is a function ϕ : R → R′ such that
1. ϕ(a + b) = ϕ(a) + ϕ(b) for all a, b ∈ R.
2. ϕ(ab) = ϕ(a)ϕ(b) for all a, b ∈ R.
Example 1. Let R be an integral domain, and let F be its field of quotients. The function ϕ : R → F
given by ϕ(a) = [a, 1] is easily seen to be a homomorphism.
Example 2. Let R be a ring, and let R[x] be its ring of polynomials. The function ϕ : R → R[x], given
by ϕ(a) = (a, 0, . . . ) = a is a homomorphism.
Example 3. Complex conjugation z = a + bi 7→ z = a − bi is a homomorphism ℂ 7→ ℂ.
Example 4. Choose m ≥ 2 and define the ring homomorphism ϕ : ℤ → ℤm by f (n) = n mod m,
that is f (n) = [n] ∈ ℤm is the congruence class of n modulo m.
– Typeset by FoilTEX –
2
Algebra course FMA190/FMA190F
2011, Spring term 2011, Sergei Silvestrov lectures
Properties of homomorphisms
Theorem 3. If ϕ : R → R′ is a ring homomorphism, then, for all a ∈ R,
1. If R has unity 1, then ϕ(1) is unity for ϕ[R].
2. ϕ(an ) = ϕ(a)n for all n ≥ 0.
3. If a is a unit, then ϕ(a) is a unit and ϕ(a−n ) = ϕ(a)−n for all n ≥ 1.
Proof.
1. For all a ∈ R,
ϕ(a) = ϕ(1a) = ϕ(a1) = ϕ(1)ϕ(a) = ϕ(a)ϕ(1),
so ϕ(1) is unity for ϕ[R].
2. Induction on n ≥ 0.
3. If ab = 1, then 1 = f (ab) = f (a) f (b), so ϕ(a−1 ) = ϕ(a)−1 . Then use induction on n ≥ 1.
– Typeset by FoilTEX –
3
Algebra course FMA190/FMA190F
2011, Spring term 2011, Sergei Silvestrov lectures
Kernels and ideals
Definition 4. The kernel of a homomorphism of rings ϕ : R → R′ is its kernel as a map of additive
groups; that is, Ker(ϕ) = ϕ −1 (0).
Definition 5. (sec 6.1, p. 135)
A subset I of a ring R is an ideal (or two-sided ideal when ring R is non-commutative) if
1. I is an additive subgroup of R, which means that it is a subset of R closed under addition in R;
2. if r ∈ R and a ∈ I , then ar ∈ I and ra ∈ I .
The equivalent reformulation of this defintion is
Definition 6. (sec 6.1, p. 135)
A subring I of a ring R is an ideal (or two-sided ideal when ring R is non-commutative) if
r ∈ R,
a∈I
⇒
ar ∈ I,
ra ∈ I.
Example 5. Two ideals of a ring R are R itself (improper ideal) and {0} (trivial ideal).
Example 6. For each integer n the cyclic subgroup nℤ is an ideal in ℤ.
Example 7. For any subset S ∈ ℝ the set of real or complex valued (continuous) functions vanishing
on S (that is f (x) = 0 for all x ∈ S) is an ideal in the ring of all (continuous) functions C(ℝ).
– Typeset by FoilTEX –
4
Algebra course FMA190/FMA190F
2011, Spring term 2011, Sergei Silvestrov lectures
A parallel with group theory
Glimpse into the future lectures on groups
Ideals play approximately the same role in the theory of rings as normal subgroups
do in the theory of groups. For instance, let R be a ring and I an ideal of R. Since the
additive group of R is abelian, I is a normal subgroup. Consequently, there is a welldefined factor group R/I in which addition is given by (a + I) + (b + I) = (a + b) + I .
R/I can in fact be made into a ring.
As one might suspect from the analogy with groups, ideals and homomorphisms of rings are
closely related. Various isomorphism theorems for groups carry over to rings with normal subgroups
and groups replaced by ideals and rings respectively. In each case the desired isomorphism is
known to exist for additive abelian groups. If the groups involved are, in fact, rings and the normal subgroups ideals, then one need only verify that the known isomorphism of groups is also a
homomorphism and hence an isomorphism of rings.
– Typeset by FoilTEX –
5
Algebra course FMA190/FMA190F
2011, Spring term 2011, Sergei Silvestrov lectures
Factor rings from homomorphisms
Theorem 4. (Th. 6.10, Sec 6.2, p 147)
The kernel of a ring homomorphism ϕ : R → R from a ring R to a ring R′
Ker(ϕ) = {r ∈ R : ϕ(r) = 0R } = ϕ −1 (0R )
is an ideal in R.
Theorem 5. (Theorem 6.9, Sec 6.2, p 147)
Let R be a ring and I an ideal of R. Then the additive factor group R/I is a ring (factor ring) with
multiplication given by
(a + I)(b + I) = ab + I.
If R is commutative or has a unity, then the same is true of R/I .
Proof. Once we have shown that multiplication in R/I is well defined, the proof that R/I is a ring
is routine. Suppose a + I = a′ + I and b + I = b′ + I . We must show that ab + I = a′ b′ + I .
Since a′ ∈ a′ + I = a + I , a′ = a + i for some i ∈ I . Similarly, b′ = b + j with j ∈ I . Consequently
a′ b′ = (a + i)(b + j) = ab + ib + a j + i j. Since I is an ideal,
a′ b′ − ab = ab + a j + i j ∈ I.
Therefore a′ b′ + I = ab + I , whence multiplication in R/I is well defined.
Example: the residue classes
Example 8 (Example revisited). Let R = ℤ, let R′ = ℤn and let γ : ℤ → ℤn maps an integer m ∈ ℤ
to the reminder γ(m) when m is divided by n. γ is a homomorphism of rings.
The kernel of γ is nℤ. By Theorem 6, the factor ring ℤ/nℤ is isomorphic to ℤn . The cosets
of nℤ are the residue classes modulo n. The isomorphism γ : ℤ/nℤ → ℤn assigns to each residue
class its smallest nonnegative element.
Theorem 6. (Theorem 6.12, Sec 6.2, p 148)
Let R be a ring and I an ideal of R. Then the map π : R → R/I given by
π(r) = r + I
is a surjective homomorphism, and its kernel is
Ker(π) = I
– Typeset by FoilTEX –
6
Algebra course FMA190/FMA190F
2011, Spring term 2011, Sergei Silvestrov lectures
Factor rings from ideals
Theorem 7. Let I be an additive subgroup of a ring R. The coset multiplication
(a + I)(b + I) = (ab) + I
is well defined, independent of the choices a and b from the cosets, and makes the group R/I of
left cosets into a ring if and only if I is an ideal of R.
The first isomorphism theorem for rings
Theorem 8. (The first isomorphism theorem for rings)
(Th. 6.13, Sec. 6.2, p 149)
If ϕ : R → R′ is a homomorphism with kernel K , then ϕ[R] is a ring, and µ : R/K → Im(ϕ) ≤ R′
given by µ(a + K) = ϕ(a) is an isomorphism. If γ : R → R/K is the homomorphism given by
γ(a) = a + K , then ϕ = µ ∘ γ .
ϕ
R
/
@@
@@
@@
@@
ϕ[R] ≤ R′
v:
vv
v
v
vv
vv
µ
γ
R/K
– Typeset by FoilTEX –
7