Let R be a ring, suppose that 2 non-zero functions f and g with:

f = a_0 + a_1x +...+ a_mx^m belongs to R[x] , a_m not equal 0 and deg(f) = m

g = b_0 + b_1x +....+ b_nx^n belongs to R[x], b_n not equal 0 and deg(g) = n

(a) Let R be and Integral Domain (ID). Show that f.g is nonzero and f.g has leading term is (a_m)(b_n)x^(m+n). Deduce that R[x] is a ID and deg(f.g) = deg(f) +deg(g). If R[x] is a ID, show that R is also a ID.

(b) Let R be a ID. Show that U(R[x]) = U(R)

(c) Let a in R and a^n = 0 for n>1. Show that (1-a) belongs to U(R). What is (1-a)^-1?

(d) Show that U(Z_4) is strictly not subset of U(Z_4[x]). Why doesn't this contradict part (b)?

Note: Z_4 is set of integer mod 4

Ok, Part (a) is easy and straightforwards.

f.g is nonzero sine both f and g are nonzero.

Also, the highest power of f.g is (a_m)(b_n)x^(m+n) obtained when u multiplied 2 functions.

R[x] is an ID because either f = 0 or g = 0 (not both equal o at the same time)

deg(f.g) = m + n = deg(f) +deg(g)

Part (b), (c) and (d) I dont know how to begin with

Can someone show me how to do part b, c and d please?

Thank you