Question

Let
F and L be fields, and

let φ : F → L be a ring homomorphism.

(a) Prove that either φ is one to one or φ is the trivial
homomorphism.

(b) Prove that if charF= charL, then φ is the trivial
homomorphism

Answer #1

Let φ : A → B be a group homomorphism. Prove that ker φ is a
normal subgroup of A.

Let
φ:G ——H be a group homomorphism and K=ker(φ). Assume that xK=yK.
Prove that φ(x)=φ(y)

Let E/F be an algebraic extension. Let K and L be intermediate
fields (i.e. F ⊆ K ⊆ E and F ⊆ L ⊆ E).
(i) Prove that if the extension K/F is separable then the
extension KL/L is separable.
(ii) Prove that if the extension K/F is normal then the
extension KL/L is normal.
Note: To make things easier for you, you can assume that E/F is
finite (hence all extensions are finite),

Let G be an Abelian group. Let k ∈ Z be nonzero. Define φ : G →
G by φ(x) = x^ k . (a) Prove that φ is a group homomorphism. (b)
Assume that G is finite and |G| is relatively prime to k. Prove
that Ker φ = {e}.

Let φ : G → G′ be an onto homomorphism and let N be a normal
subgroup of G. Prove that φ(N) is a normal subgroup of G′.

Prove the following theorem: Let φ: G→G′ be a group
homomorphism, and let H=ker(φ). Let a∈G.Then the set
(φ)^{-1}[{φ(a)}] ={x∈G|φ(x)} =φ(a)
is the left coset aH of H, and is also the right coset Ha of H.
Consequently, the two partitions of G into left cosets and into
right cosets of H are the same

Let R be a ring. Show that the set Aut(R) = {φ : R → R|φ is a
ring isomorphism} is a group with composition.

Letφ:G→G′is a group homomorphism. Prove that φ is one-to-one if
and only if Ker(φ) ={e}.

Let I be an ideal of the ring R. Prove that the reduction map
R[x] → (R/I)[x] is a ring homomorphism.

Let
R be a ring, and let N be an ideal of R.
Let γ : R → R/N be the canonical homomorphism.
(a) Let I be an ideal of R such that I ⊇ N.
Prove that γ−1[γ[I]] = I.
(b) Prove that mapping
{ideals I of R such that I ⊇ N} −→ {ideals of R/N} is a
well-defined bijection between two sets

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