Do you know what a ring/subring is? If so just check the definitions. Do you need help with checking which conditions hold and which fails? You should post some of your work because this is a long problem.
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 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?