Assignment 9 – Due Friday, November 14
Turn this in at the start of recitation on Friday, November 14.
1. (Judson, p. 240, #1) Which of the following sets are rings with respect to the usual operations of
addition and multiplication? If the set is a ring, is it also a field?
(a) 4Z
(b) Z≥0 = {z ∈ Z | z ≥ 0}
(c) Z20
√
√
(d) Q( 2) = {a + b 2 | a, b ∈ Q}
√
(e) R = {a + b 3 2 | a, b ∈ Q}
2. Let
R = {f : [0, 1] → R | f is continuous}
be the set of continuous real-valued functions on the interval [0, 1]. Show that R is a ring with respect
to the operations (f + g)(x) = f (x) + g(x) and (f g)(x) = f (x)g(x). Show that R is a unital ring
(i.e., has a multiplicative identity). What elements of R have multiplicative inverses? Is R an integral
domain?
3. If R is a ring, n ∈ N, and a ∈ R, let
na = a + · · · + a .
| {z }
n times
Show that if a, b ∈ R and m, n ∈ N, then
(na)(mb) = (nm)(ab).
4. Herstein, p. 130: #3 (see #2 for the n = 2 case), #4
1