Download Math 3101 Spring 2017 Homework 2 1. Let R be a unital ring and let

Math 3101
Spring 2017
Homework 2
1. Let R be a unital ring and let I be a left ideal of R. Show that I = R if and only
if I contains 1R . (Note that any ring is in fact a left, right, and two-sided ideal of
itself; you do not have to prove this fact. Also note that the statement in the problem
remains true if I is instead assumed to be a right ideal or a two-sided ideal.)
2. Let R be a unital ring, let n be a positive integer, and let Mn (R) denote the unital
ring of n×n matrices with entries in R. (You do not have to prove that Mn (R) is a
unital ring.)
(a) Let I be an ideal of R. Prove that
a b Mn (I) =
a, b, c, d ∈ I
c d is an ideal of Mn (R).
(b) Let J be an ideal of Mn (R), and let E(J) denote the subset of R comprised of
entries of matrices in J. Prove that E(J) is an ideal of R, and that J = Mn (E(J)).
(Hint: Consider the elementary matrices eij for 1 ≤ i, j ≤ n.) Conclude that every
ideal of Mn (R) has the form Mn (I) for some choice of ideal I of R.
(c) Let K be a field. Conclude from the above that the only ideals of Mn (K) are
the zero ideal and Mn (K) itself.
3. Let K be a field, and let R be the ring of 2×2 matrices with entries in K. Find a
left ideal of R that is not a right ideal and a right ideal of R that is not a left ideal.
Justify your answers.
4. Let K be a field, and let R be the set of upper triangular 2×2 matrices with
entries in K. That is, R is comprised of all matrices of the form
a b
0 c
for a, b, c ∈ K.
(a) Prove that R is a unital subring of M2 (K).
(b) Find five distinct two-sided ideals of R. Justify your answer. (Can you prove
that this list is complete?)
5. Let X be a nonempty set and let R be a ring. Recall the set F (X, R) of functions
from X to R, and recall that F (X, R) is a ring under pointwise addition and multiplication. (You do not have to prove that F (X, R) is a ring.) Note, for every r ∈ R,
that we have the constant function fr with fr (x) = r for all x ∈ X.
(a) Prove that the set of constant functions in F (X, R) forms a subring of F (X, R).
(b) Prove that the subring of constant functions of F (X, R) is isomorphic to R
itself.