Download Problem set 7

MATH 210
Problem Set 7
Due: March 4, 2016
Exercises
1. Let f : R → S be a homomorphism of commutative rings.
(a) For any element r ∈ R and any positive integer n, show that
f (nr) = nf (r)
and
f (rn ) = f (r)n .
(b) If I ⊂ R is an ideal of R, and f is surjective, then prove that f (I) is an ideal of S.
(c) Show that f is an isomorphism if and only if ker f = h0i and f (R) = S.
2. (a) Which of the following maps from C to C are ring homomorphisms?
(i) x + yi 7→ x,
(ii) x + yi 7→ x − yi,
(iii) x + yi 7→ |x + yi|2 = x2 + y 2 ,
(iv) x + yi 7→ y + xi.
(b) In the ring of integers, find a positive integer n such that
hni = h2i + h3i.
(c) In the ring of integers, find a positive integer m such that
hmi = hai + hbi
for any a, b ∈ Z.
3. To try to get comfortable with abstract rings, let’s compare the cases we have seen to the ‘more
abstract’ and see how similar they can be.
(a) Show that Z/h6i is not a field.
(b) Show that (Z/3Z)[x]/hx2 + x + 1i is not a field.
(c) Let F be the ring of continuous real-valued functions in one variable, with pointwise addition and
multiplication, i.e., for f (x), g(x) ∈ F we have (f + g)(x) = f (x) + g(x) and (f g)(x) = f (x)g(x).
Let I be the collection of functions f (x) in F with f (1/3) = f (2/3) = 0. Show that I is an ideal
and that
F/I
is not a field.
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Problems
4. Let I and J be ideals in a commutative ring R.
(a) Prove that IJ ⊂ I ∩ J.
(b) If I + J = R, show that I ∩ J = IJ.
(c) Give an example of two ideals I, J ⊂ Z such that I ∩ J 6= IJ.
5. Let R be a commutative ring, and let I ⊂ R be an ideal.
(a) Show that the nil radical of I,
N (I) = {r ∈ R | rn ∈ I for some positive integer n}
is an ideal of R.
(b) Show that R/N (h0i) has no nonzero nilpotent elements. (N (h0i) is called the nil radical of R.)
(c) Suppose I is an ideal in a ring J, and J is an ideal in a ring R. Show that if I contains the
multiplicative identity, then I is also an ideal of R.
Food for thought
A field is a very nice (commutative) ring, it is a ring where all non-zero elements are units, i.e., all but one
element is a unit. What about commutative rings where all but two elements are units. Can you classify
the possibilities, up to isomorphism?
Practice problems
(1) Let
a
R=
b
2b
a
: a, b ∈ R .
√
Show that R and Z[ 2] are isomorphic as rings. (See Shifrin 4.1.11.)
(2) Let I, J ⊂ R be two ideals in a commutative ring R. Show that I ∪ J is an ideal if and only if I ⊂ J
or J ⊂ I.
(3) Let I = hx2 + x + 2i ⊂ (Z/5Z)[x]. Find the multiplicative inverse of 2x + 3 + I in (Z/5Z)[x]/I.
(4) Shifrin section 4.1: 1, 2, 3, 4, 5, 7, 11, 17.
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