Let R be a ring with unity 1, and S = {n.1 : n E Z} . Then S'is Ra subring of Rnot a subring of

. Let R = Z. Find 5 examples of principal ideals. . Let R = R[x]. Find 5 examples of principal ideals . Consider the ring (P(Z), Δ,∩). What is ⟨{2}⟩?

6. (10%) Let R be the ring of R of all real numbers of the form x+yv2 with respect to standard addition and multiplication, where r and y are integers, and let T be the subset of T of the numbers of the form m + ny2, where m is an even integer, and n is an integer. Prove that T is an ideal of R.

Consider the ring R = 2Z and the ideal I = 24Z of R. Give a representative for each coset of I in R.