Logo
Home Study Material Contact Search
AI Chat FREE
  1. mathematics
  2. advance math
  3. 9.2.6 exercise. let r = z and let i be the ideal 1

9.2.6 Exercise. Let R = Z and let I be the ideal 12Z of R. (i)...

80.2K

Verified Solution

Question

Advance Math

9.2.6 Exercise. Let R = Z and let I be the ideal 12Z of R.

  1. (i) List explicitly all the ideals A of R with I ? A.
  2. (ii) Write out all the elements of R/I (these are cosets).
  3. (iii) List explicitly the set of all ideals B of R/I (these aresets of cosets).
  4. (iv) Let ?: R ? R/I be the natural projection. For each ideal Aof R such that I ? A, write out ?(A) explicitly (this is a set ofcosets). Confirm by direct calculation what the Third IsomorphismTheorem says: that the function A ?? ?(A) is a bijection from theset of such A that you found in step (i) and the set of ideals ofR/I that you found in step (iii).
  5. (v) For each ideal A of R with I ? A, write out all theelements of the following three quotient groups (under addition):R/A, A/I, and (R/I)/(A/I) (the last consists of cosets of cosets!).Then confirm by direct calculation what the Third IsomorphismTheorem says: that the rule f : R/A ? (R/I)/(A/I) with f(gA) =(gI)(A/I) makes a well defined function that is an isomorphism ofrings.

Answer & Explanation Solved by verified expert
3.7 Ratings (567 Votes)
    See Answer
Get Answers to Unlimited Questions

Join us to gain access to millions of questions and expert answers. Enjoy exclusive benefits tailored just for you!

Membership Benefits:
  • Unlimited Question Access with detailed Answers
  • Zin AI - 3 Million Words
  • 10 Dall-E 3 Images
  • 20 Plot Generations
  • Conversation with Dialogue Memory
  • No Ads, Ever!
  • Access to Our Best AI Platform: Flex AI - Your personal assistant for all your inquiries!
Become a Member

Other questions asked by students

Q

a.)Using disks or washers, find the volume of the solid obtained by rotating the region bounded...

Basic Math
Q

A random sample of ?n measurements was selected from a population with standard deviation ?=11.7 and...

Statistics
Q

a. A student performed Part 3 of the lab. A 360 ppm Ca2+ standard required 22.34...

Chemistry
Q

1) a) Compare pin photodiode with pn junction photodiode. Please write one application best for each...

Physics
Q

Keller Construction is considering two new investments. Project E calls for the purchase of earthmoving equipment....

Accounting
Q

Matter Example 5 5 Figure 5 5 shows a small magnetised needle P placed at...

Physics
Q

True or false 1 4 of 0 is still 0 True False

Basic Math
Q

compare and contrast financial accounting with management accounting. specify at least three (3) areas where...

Accounting
Q

Which of the following items is reported on the income statement as an expense? This...

Accounting
Q

How do we calculate the income tax value for bad debts in the above question....

Accounting
Q

which of the following statements referring to a production report is not correct: a: the...

Accounting
Q

Morrisey & Brown, Ltd., of Sydney is a merchandising company that is the sole distributor...

Accounting