Can someone please help me with these questions. I attached them below.
Thanks.
I wish I could show my work on here, but I'm not too adept in writing in that LaTex code you use. For question 4, part a, I went through all the steps, and assuming everything I did is correct, I concluded that this was a commutative ring. Now, I'm not 100% sure for the identity part of the question. I know zero is the additive identity, but is 1 the multiplicative identity? I'm also not sure how to find the units for question 4, part a. By definition, I need to find all the elements that when multiplied by {a + b*root(d)} give one correct? I'm not sure how to find such elements. Maybe there aren't any?
yes, there aren't any..
what you need to get the identity is of the form $\displaystyle \frac{1}{a+b\sqrt d} = \frac{a - b\sqrt d}{a^2 + b^2d}$ which is not in the set..
i assume, 4b would be easy.. it is not a ring.. in fact, it is not a group under addition..
for 5a, what does the bar above the numbers mean?
anyways, it can easily be shown that if a,b in S, then a-b is in S and ab in S.. which will prove that S is a subring..
OK, so I nearly have everything done now. I'm just stuck on the very last question (5b). I'm fairly certain that the set S of matrices is a commutative subring of R (please correct me if I'm wrong though). Now, I'm just having problems finding the identity and units of S. Does anyone have any ideas?