3. Consider the function f: ZQ defined by f(n) = n. a) Prove f is a ring homomorphism. b) Find the image of f. Recall, im f = {q EQ | 3n e Z such that f(n) = q}. c) Is im f an ideal in Q? If yes, prove it. If no, provide a counterexample. d) Show that im f is closed under addition and multiplication. e) Look at what you have done in (c) and (d). Fill in the blanks based on your answers. Conjecture: im f is NOT an in Q but im f is a of Q.
3. Consider the function f: ZQ defined by f(n) = n. a) Prove f is a ring homomorphism. b) Find the image of f. Recall, im f = {q EQ | 3n e Z such that f(n) = q}. c) Is im f an ideal in Q? If yes, prove it. If no, provide a counterexample. d) Show that im f is closed under addition and multiplication. e) Look at what you have done in (c) and (d). Fill in the blanks based on your answers. Conjecture: im f is NOT an in Q but im f is a of Q.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.1: Polynomials Over A Ring
Problem 17E
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Transcribed Image Text:3.
Consider the function f: ZQ defined by f(n) = n.
a) Prove f is a ring homomorphism.
b) Find the image of f. Recall, im f = {q EQ | 3n e Z such that f(n) = q}.
c) Is im f an ideal in Q? If yes, prove it. If no, provide a counterexample.
d) Show that im f is closed under addition and multiplication.
e) Look at what you have done in (c) and (d). Fill in the blanks based on your answers.
Conjecture: im f is NOT an
in Q but im f is a
of
Q.
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