Math 150b: Modern Algebra
Homework 7
This problem set is due Friday, February 19. Do problems 11.3.5(a), 11.3.10, 11.3.13, 11.4.3(b), 11.5.2,
and 11.5.3, in addition to my problems below.
Remarks on problem 11.3.13: The notation IJ is a deviation from strict set arithmetic, since in that
interpretation it would only be the set of products {xy}. In this strict notation, the product ideal would
be written (IJ), the ideal generated by the set product IJ. Also, the problem asks for an example where
IJ 6= (IJ). Problem GK7.2 provides a hint: don’t use principal ideals.
This time, several of the book problems assume that the rings used are commutative.
G7.1. In class I said rather informally that a category is a collection of “favorite” objects (sets, groups, rings,
etc.) and a collection of allowed maps (all functions, group homomorphisms, ring homomorphisms),
etc. A more precise definition includes these conditions: (1) the identity map on every object is
allowed, and (2) the composition of two allowed maps is allowed. I also said that an initial object in a
category is one that has a unique allowed map to any other object; a final object has a unique allowed
map from every other object. The book shows that Z is an initial object in the category of rings. In
class, we also discussed these examples: The empty set is an initial set. A one-element set is a final
set. The trivial group is an initial and final group. The trivial ring (in which 1 = 0) is a final ring.
(a) We can consider the category of fields, where field homomorphisms are exactly ring homomorphisms: functions that preserve addition and multiplication (and take 1 to 1). Show that all field
homomorphisms are injective. (Hint: There is a relevant proposition in the book.) Are they all
surjective?
(b) Show that F p is an initial field in the category of fields of characteristic p, and that Q is an initial
field in the category of fields of characteristic 0. Show that there is no initial field the category
of all fields (i.e., all characteristics).
*(c) Show that there is no final object in the category of fields of characteristic p, nor the category of
fields of characteristic 0.
GK7.2. Let I, J ⊆ R be two ideals in a (commutative) ring. Show that if I is principal, then the set arithmetic
product
IJ = {xy|x ∈ I, y ∈ J}
is the same set as the product ideal (IJ).
GK7.3. Let F be a field, and consider the ring F[ε] ∼
= F[x]/x2 mentioned in class, where the symbol ε
2
satisfies ε = 0. and let f (x) be a polynomial with coefficients in F. Prove that
f (a + ε) = f (a) + ε f 0 (a),
where f 0 (x) is as defined in problem 11.3.5.
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