Math 653 Homework Assignment 8
1. (a) Let R and S be rings with 1 6= 0. Prove that every ideal of R × S is of the
form I × J where I is an ideal of R and J is an ideal of S.
(b) Give an example of groups G and H for which there is a normal subgroup
of G × H that is not of the form A × B for any normal subgroup A of G
and normal subgroup B of H.
2. An element e in a ring R is idempotent if e2 = e. Suppose e is idempotent and
central (that is er = re for all r ∈ R).
(a) Prove that Re and R(1 − e) are (two-sided) ideals of R.
(b) Show that Re and R(1−e) are rings with identities e and 1−e, respectively,
and that R ∼
= Re × R(1 − e).
3. Let R be a ring with 1 6= 0. Prove that R is a division ring if, and only if, the
only left ideals of R are 0 and R.
4. Let U denote the subring of M2 (R) consisting of upper triangular matrices, and
let I denote the ideal of U consisting of strictly upper triangular matrices. That
is,
0 b
a b
|b∈R .
| a, b, c ∈ R
and I =
U=
0 0
0 c
Use the First Isomorphism Theorem to show that U/I ∼
= R × R as rings.
5. Let f : R → S be a homomorphism of commutative rings. Let P be a prime
ideal and M a maximal ideal of S.
(a) Prove that f −1 (P ) is a prime ideal of R.
(b) If R is a subring of S, and f is the inclusion homomorphism, use (a) to
prove that P ∩ R is a prime ideal of R.
(c) Prove that if f is surjective, then f −1 (M ) is a maximal ideal of R. Give an
example to show that this may not be true if f is not surjective.
6. Solve Sun Zi’s problem: “We have a number of things, but we do not know
exactly how many. If we count them by threes we have 2 left over. If we count
them by fives we have 3 left over. If we count them by sevens we have 2 left over.
How many things are there?” In our notation, this means to solve the following
system of congruences for x ∈ Z.
x ≡ 2 (mod 3)
x ≡ 3 (mod 5)
x ≡ 2 (mod 7)
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