Other
MAT3702 Assignment TWO 2023
- Course
- Institution
UNISA Abstract Algebra MAT3702 Assignment TWO of 2023 solutions.
[Show more]
Seller
Follow
lyzo2005
Reviews received
Content preview
MAT3702 ASSIGNMENT TWO 2023Question 1
𝑎𝑝−1 ≡ 24 (𝑚𝑜𝑑 5)
𝑎𝑝−1 ≡ 16 (𝑚𝑜𝑑 5)
15 = 5 × 3
16 − 1 = 5 × 3
5 divides (16 − 1)
Therefore,
𝑎𝑝−1 ≡ 1 (𝑚𝑜𝑑 5)
Question 2
𝑆 = {0, 3} is not a subring of ℤ6 because 𝑆 does not contain 1.
Question 3
𝑅 is a ring and 𝑏 ∈ 𝑅 such that 𝑏𝑏 = 𝑏.
Given that 𝑏 ≠ 0𝑅
Therefore, 𝑏 has a unique inverse 𝑏 −1 ∈ 𝑅 such that 𝑏𝑏 −1 = 1𝑅 = 𝑏 −1 𝑏
𝑏𝑏 = 𝑏
⇒ (𝑏𝑏)𝑏 −1 = 𝑏𝑏 −1
Rings are associative, therefore:
𝑏(𝑏𝑏 −1 ) = 𝑏𝑏 −1
𝑏1𝑅 = 1𝑅
𝑏 = 1𝑅
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller lyzo2005. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $11.04. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
52355 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now