Download Chapter 14 Ideals and Factor Rings Definition A subring A of a ring

Chapter 14 Ideals and Factor Rings
Definition A subring A of a ring R is a (two-sided) ideal if ar, ra ∈ A for every r ∈ R and every
a ∈ A.
[An ideal is a subring with left and right absorbing power!]
Theorem 14.1 A non-empty subset A of a ring R is an ideal if
1. a − b ∈ A whenever a, b ∈ A, and
2. ra, ar ∈ A whenever a ∈ A, r ∈ R.
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Example {0}, nZ in Z, hf (x)i in R[x], h{2, x}i = (2Z)[x] ⊆ Z[x].
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Theorem 14.2 Let R be a ring, and A be a subring. The set of cosets {r + A : r ∈ R} is a ring
under the operations
(x + A) + (y + A) = (x + y) + A and (x + A)(y + A) = xy + A
is a ring (known as the factor ring) if and only if A is an ideal.
Proof. Key step is to show that the multiplication is well-defined if and only if A is an ideal.
If A is an ideal, then ...
If there are a ∈ A, r ∈ R such that ar ∈
/ A, then ...
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Examples Z/4Z, 2Z/6Z, Z[i]/h2 − ii, R[x]/hx2 + 1i.
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Definition Let A be a proper ideal of a commutative ring R.
It is a prime ideal a, b ∈ R satisfying ab ∈ A imply that a ∈ A or b ∈ A.
It is a maximal ideal if there is no other ideal lying strictly between A and R.
Example nZ is prime if and only if n is prime; h2i and h3i are maximal ideal in Z36 ; hx2 + 1i is a
maximal ideal in R[x].
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Theorem 14.3-4 Let A be an ideal of a commutative ring R with unity.
(a) The factor ring R/A is an integral domain if and only if A is prime.
(b) The factor ring R/A is a field if and only if A is maximal.
Remark If A is maximal, then A is prime. The ideal hxi is prime in Z[x], but not maximal.
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Chapter 15 Ring Homomorphisms
Definitions Let R1 , R2 be rings. A function φ : R1 → R2 is a ring homomorphism if φ(a + b) =
φ(a) + φ(b) and φ(ab) = φ(a)φ(b) for all a, b ∈ R1 . If in addition that φ is bijective, then φ is a ring
isomorphism.
Theorem 15.1-2 let φ : R1 → R2 be a ring homomoprhism.
(1) For any r ∈ R and positive integer n, φ(nr) = nφ(r) and φ(rn ) = φ(r)n .
(2) If A is a subring of R1 , then φ(A) is a subring of R2 .
(3) If A is an ideal of R1 , then φ(A) is an ideal of φ(R1 ).
(4) If B is a subring/ideal of R2 , then φ−1 (B) is a subring/ideal of R1 . In particular, Ker(φ) is
an ideal.
(5) If A is a commutative subring of R, then φ(A) is commutative.
(6) If R1 has a unity, then φ(1) is a unity of φ(R1 ).
(7) The map φ is injective if and only if Ker(φ) = {0},
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Theorem 15.3 Let φ : R1 → R2 be a ring homomorphism. Then x + Ker(φ) 7→ φ(x) is an
isomorphism from R1 /Ker(φ) to φ(R1 ).
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Theorem 15.4 Every ideal A of a ring R is the kernel of the ring homomorphism φ : A 7→ R/A
defined by φ(a) = a + A.
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Theorem 15.5 Let R be a ring with unity. Then φ : Z → R defined by φ(n) = n · 1 is a ring
homomorphism. In particular, φ(Z) = Z or Zn . In the former case, R has characteristic 0; in the
latter case R has characteristic n.
Corollary (a) φ : Z → Zn defined by φ(a) = [a] is a ring homomorphism.
(b) If R is a field of characteristic p, then {n · 1 : n ∈ Z} is isomorphic to Zp .
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Construction of the fields of quotients
Theorem 15.6 Let D be an integral domain. Then (a, b) ∼ (c, d) on D × D∗ defined by ad = bc
is an equivalence relation. Suppose F = {[(a, b)] : (a, b) ∈ D × D∗ } is the set of equivalence
classes of the relation. Then F is a field under the operations [(a, b)] + [(c, d)] = [(ad + bc, bd)] and
[(a, b)][(c, d)] = [(ab, cd)]. This is known as the field of quotients of D.
Examples D = Z, Z[x], Zp [x] for a prime p, and R[x].
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Construction of finite field
Theorem Suppose F is field, and A = hf (x)i ∈ F[x] is a maximal ideal, where
f (x) = xm + am−1 xm−1 + · · · + a0 .
Then A = {f (x)q(x) : q(x) ∈ F[2]}, and F[x]/A = {r(x) + A : r(x) ∈ F[x] of degree at most m − 1}
is a field. If F = Zp and f (x), then F[x]/A has pm elements.
Question When is A = hf (x)i maximal if f (x) ∈ F[x]?
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