Question

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.

c) Is I maximal in R? Prove your claim.

(HINT: For maximality, do not try to prove/disprove maximality directly. Instead, think about properties of maximal ideals from class that must hold true and apply them to this ideal.)

Answer #1

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 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 ·...

Given that A1 = B1 minus B2,
A2 = B2 minus B3, and
A3 = B3 minus B1, Find the joint
p.m.f. (probability mass function) of A1 and
A2, where Ai ~ Ber(p) for all random
variables i in {1,2,3}

Given that A1 = B1 minus
B2,A2 =
B2 minus B3, and
A3 = B3 minus
B1, Find the joint p.m.f.
(probability mass function) of A1 and A2,
where Bi ~ Ber(p) for all
random variables i in {1,2,3}

Let A = (A1, A2, A3,.....Ai) be defined as a sequence containing
positive and negative integer numbers.
A substring is defined as (An, An+1,.....Am) where 1 <= n
< m <= i.
Now, the weight of the substring is the sum of all its
elements.
Showing your algorithms and proper working:
1) Does there exist a substring with no weight or zero
weight?
2) Please list the substring which contains the maximum weight
found in the sequence.

Prove Theorem 29.10. Let n ∈ Z+. If Ai is countable for all i =
1,2,...,n, then A1 ×A2 ×···×An is countable.

Let S = {(a1,a2,...,an)|n ≥ 1,ai ∈ Z≥0 for i = 1,2,...,n,an ̸=
0}. So S is the set of all finite ordered n-tuples of nonnegative
integers where the last coordinate is not 0. Find a bijection from
S to Z+.

As per The Economist (June 24, 2017), the Argentinian
government issued its first 100‐year bond, with cash flows
denominated in dollars. The bond thus now matures in,
for simplification purposes, 97 years. The current bond has a
$1,000 face value and the following monthly, end‐ofmonth
coupon payments: $10/ month for 47 years, $30/month for 20 years,
and then $50/month for 30 years. As
Argentina has defaulted on its bonds six times in the past 100
years, you decide that a...

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