) In the Extra Problem on PS 2 we defined a map φ:M2,2(Z) →
M2,2(Z2) by...
) In the Extra Problem on PS 2 we defined a map φ:M2,2(Z) →
M2,2(Z2) by the formula φ a b c d = a mod (2) b mod (2) c mod (2) d
mod (2)
On PS 2 you showed that φ is a ring homomorphism and that ker(φ)
= 2a 2b 2c 2d a, b, c, d ∈ Z We know the kernel of any ring
homomorphism is an ideal. Let I = ker(φ).
(a) (6 points) The...
Consider the relation R defined on the set R as follows: ∀x, y ∈
R, (x,...
Consider the relation R defined on the set R as follows: ∀x, y ∈
R, (x, y) ∈ R if and only if x + 2 > y.
For example, (4, 3) is in R because 4 + 2 = 6, which is greater
than 3.
(a) Is the relation reflexive? Prove or disprove.
(b) Is the relation symmetric? Prove or disprove.
(c) Is the relation transitive? Prove or disprove.
(d) Is it an equivalence relation? Explain.
4. (30) Let C be the ring of complex numbers,and letf:C→C be the
map defined by...
4. (30) Let C be the ring of complex numbers,and letf:C→C be the
map defined by
f(z) = z^3.
(i) Prove that f is not a homomorphism of rings, by finding an
explicit counterex-
ample.
(ii) Prove that f is not injective.
(iii) Prove that the principal ideal I = 〈x^2 + x + 1〉 is not a
prime ideal of C[x].
(iv) Determine whether or not the ring C[x]/I is a field.
Prove or disprove: GL2(R), the set of invertible 2x2 matrices,
with operations of matrix addition and...
Prove or disprove: GL2(R), the set of invertible 2x2 matrices,
with operations of matrix addition and matrix multiplication is a
ring.
Prove or disprove: (Z5,+, .), the set of invertible
2x2 matrices, with operations of matrix addition and matrix
multiplication is a ring.
Let
α = 4√ 3 (∈ R), and
consider the homomorphism
ψα : Q[x] → R...
Let
α = 4√ 3 (∈ R), and
consider the homomorphism
ψα : Q[x] → R
f(x) → f(α).
(a) Prove that irr(α, Q) = x^4 −3
(b) Prove that Ker(ψα) = <x^4 −3>
(c) By applying the Fundamental Homomorphism Theorem to ψα,
prove that
L ={a0+a1α+a2α2+a3α3 | a0, a1, a2, a3 ∈ Q }is the smallest
subfield of R containing α.
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It...
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It turns out that R forms a ring under the operations:
(a1,a2,a3,···)+(b1,b2,b3,···)=(a1 +b1,a2 +b2,a3 +b3,···),
(a1,a2,a3,···)·(b1,b2,b3,···)=(a1 ·b1,a2 ·b2,a3 ·b3,···)
Let I = {(a1,a2,a3,···) ∈ Z∞ : all but finitely many ai are 0}.
You may use without proof the fact that I forms an ideal of R.
a) Is I principal in R? Prove your claim.
b) Is I prime in R? Prove your claim....
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It...
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It turns out that R forms a ring under the operations:
(a1,a2,a3,···)+(b1,b2,b3,···)=(a1 +b1,a2 +b2,a3 +b3,···),
(a1,a2,a3,···)·(b1,b2,b3,···)=(a1 ·b1,a2 ·b2,a3 ·b3,···)
Let I = {(a1,a2,a3,···) ∈ Z∞ : all but finitely many ai are 0}.
You may use without proof the fact that I forms an ideal of R.
a) Is I principal in R? Prove your claim.
b) Is I prime in R? Prove your claim....