Elements Of Modern Algebra
8th Edition
ISBN: 9781285463230
Author: Gilbert, Linda, Jimmie
Publisher: Cengage Learning,
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Chapter 7.2, Problem 19E
To determine
To prove:
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Let R be a ring with unity e. Verify that the mapping θ: Z---------- R defined by θ (x) = x • e is a homomorphism
Show that if R, R’, and R’’ are rings, and if ø : R → R’ and ψ : R’ → R’’ are homomorphisms, then the composite function ψ ø : R → R’’ is a homomorphism.
a) Let Ø. R[x] R be a map given by Øf(x)) = f(1) Show that Ø is a ring
homomorphism.
b) Describe the kernel of the map Ø in part(a).
Chapter 7 Solutions
Elements Of Modern Algebra
Ch. 7.1 - Label each of the following statements as either...Ch. 7.1 - Label each of the following statements as either...Ch. 7.1 - Label each of the following statements as either...Ch. 7.1 - Label each of the following statements as either...Ch. 7.1 - Label each of the following statements as either...Ch. 7.1 - Label each of the following statements as either...Ch. 7.1 - Label each of the following statements as either...Ch. 7.1 - Label each of the following statements as either...Ch. 7.1 - Label each of the following statements as either...Ch. 7.1 - Find the decimal representation for each of the...
Ch. 7.1 - Prob. 2ECh. 7.1 - Prob. 3ECh. 7.1 - Find the decimal representation for each of the...Ch. 7.1 - Prob. 5ECh. 7.1 - Prob. 6ECh. 7.1 - Prob. 7ECh. 7.1 - Prob. 8ECh. 7.1 - Express each of the numbers in Exercises 7-12 as a...Ch. 7.1 - Express each of the numbers in Exercises 7-12 as a...Ch. 7.1 - Express each of the numbers in Exercises 7-12 as a...Ch. 7.1 - Express each of the numbers in Exercises 7-12 as a...Ch. 7.1 - Prove that is irrational. (That is, prove there...Ch. 7.1 - Prove that is irrational.
Ch. 7.1 - Prove that if is a prime integer, then is...Ch. 7.1 - Prove that if a is rational and b is irrational,...Ch. 7.1 - Prove that if is a nonzero rational number and ...Ch. 7.1 - Prove that if is an irrational number, then is...Ch. 7.1 - Prove that if is a nonzero rational number and ...Ch. 7.1 - Give counterexamples for the following...Ch. 7.1 - Let S be a nonempty subset of an order field F....Ch. 7.1 - Prove that if F is an ordered field with F+ as its...Ch. 7.1 - If F is an ordered field, prove that F contains a...Ch. 7.1 - Prove that any ordered field must contain a...Ch. 7.1 - If and are positive real numbers, prove that...Ch. 7.1 - Prove that if and are real numbers such that ,...Ch. 7.2 - True or False
Label each of the following...Ch. 7.2 - Prob. 2TFECh. 7.2 - Prob. 3TFECh. 7.2 - True or False
Label each of the following...Ch. 7.2 - Prob. 5TFECh. 7.2 - True or False
Label each of the following...Ch. 7.2 - Prob. 7TFECh. 7.2 - Prob. 1ECh. 7.2 - Prob. 2ECh. 7.2 - Prob. 3ECh. 7.2 - Prob. 4ECh. 7.2 - Prob. 5ECh. 7.2 - Prob. 6ECh. 7.2 - Prob. 7ECh. 7.2 - Prob. 8ECh. 7.2 - Prob. 9ECh. 7.2 - Prob. 10ECh. 7.2 - Prob. 11ECh. 7.2 - Prob. 12ECh. 7.2 - Prob. 13ECh. 7.2 - Prob. 14ECh. 7.2 - Prob. 15ECh. 7.2 - Prob. 16ECh. 7.2 - Prob. 17ECh. 7.2 - Prob. 18ECh. 7.2 - Prob. 19ECh. 7.2 - Prob. 20ECh. 7.2 - Prob. 21ECh. 7.2 - Prob. 22ECh. 7.2 - Prob. 23ECh. 7.2 - Prob. 24ECh. 7.2 - Prob. 25ECh. 7.2 - Prob. 26ECh. 7.2 - Prob. 27ECh. 7.2 - Prob. 28ECh. 7.2 - Prob. 29ECh. 7.2 - Prob. 30ECh. 7.2 - Prob. 31ECh. 7.2 - Prob. 32ECh. 7.2 - Prob. 33ECh. 7.2 - Prob. 34ECh. 7.2 - Prob. 35ECh. 7.2 - Prob. 36ECh. 7.2 - Prob. 37ECh. 7.2 - Prob. 38ECh. 7.2 - Prob. 39ECh. 7.2 - Prob. 40ECh. 7.2 - Exercise are stated using the notation in the...Ch. 7.2 - Prob. 42ECh. 7.2 - Prob. 43ECh. 7.2 - Prob. 44ECh. 7.2 - Prob. 45ECh. 7.2 - Prob. 46ECh. 7.2 - Prob. 47ECh. 7.2 - Prob. 48ECh. 7.2 - Prob. 49ECh. 7.2 - Prob. 50ECh. 7.2 - An element in a ring is idempotent if . Prove...Ch. 7.2 - Prove that a finite ring R with unity and no zero...Ch. 7.3 - True or False
Label each of the following...Ch. 7.3 - Prob. 2TFECh. 7.3 - Prob. 3TFECh. 7.3 - Prob. 4TFECh. 7.3 - Prob. 1ECh. 7.3 - Find each of the following products. Write each...Ch. 7.3 - Prob. 3ECh. 7.3 - Show that the n distinct n th roots of 1 are...Ch. 7.3 - Prob. 5ECh. 7.3 - Prob. 6ECh. 7.3 - Prob. 7ECh. 7.3 - Prob. 8ECh. 7.3 - Prob. 9ECh. 7.3 - Prob. 10ECh. 7.3 - Prob. 11ECh. 7.3 - Prob. 12ECh. 7.3 - Prob. 13ECh. 7.3 - Prob. 14ECh. 7.3 - Prove that the group in Exercise is cyclic, with ...Ch. 7.3 - Prob. 16ECh. 7.3 - Prob. 17ECh. 7.3 - Prob. 18ECh. 7.3 - Prob. 19ECh. 7.3 - Prob. 20ECh. 7.3 - Prob. 21ECh. 7.3 - Prob. 22ECh. 7.3 - Prove that the set of all complex numbers that...Ch. 7.3 - Prob. 24ECh. 7.3 - Prob. 25ECh. 7.3 - Prob. 26ECh. 7.3 - Prob. 27ECh. 7.3 - Prob. 28E
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- Consider the ring homomorphism : I→ R defined as (a) Show that is a ring homomorphism. (b) Use FIT for rings to show that I/JR as rings. y × ( [ ; ² ]) = - y =x-Y.arrow_forwardLet F=C be the field of complex numbers and let V be any space over C . Suppose that f and g are linear functional on V such that the function h (v) = f (v) g (v) is also a functional linear over V. Show that f = 0 or g = 0.arrow_forwardGive an example of ring elements a and b with the proerties that ab = 0 but ba does not equal 0.arrow_forward
- (k) Let f: A B be a mapping of sets, A a system of subsets of A, and B a system of subsets of B. Define S(A) = {S(X) C B : X e A). (8) = {r-)eA :Y e B}. Prove that f-1(B) is a ring if B is a ring. (1) Show that f(A) is not necessarily a ring if A is a ring. (m) Show that -(B) is a a-algebra if B is a o-algebra. (n) Prove that R(s-(8) = (R(B). where R(C) is the minimal ring containing e.arrow_forwardExplain why the polynomial rings R[r] and C[x] are not isomorphic.arrow_forwardCheck whether the function f: A → B is or is not a homomorphism of ring A in ring B in the following cases: a) A = Z, B = Z, f (x) = x + 1 b) A = Z, B = Z, f (x) = 2xarrow_forward
- [a 2b Let Z[2] = {a +b2 |a, b e Z} and let H= { |b : a,b eZ }. a Show that Z[/2] and H are isomorphic as rings. 2.arrow_forwardConsider ϕ: ℤ ×ℤ → ℤ defined by ϕ(a,b) = a-b. Prove that ϕ is a homomorphism.arrow_forwardB E None (i), (ii) (iii), (iv) Which of the following are ring homomorphisms? (i): f: (Q(√2), +,-) → (Q(√3), +,-), a +b√2+a+b√3. (ii): f: C Rx R, a +bi (a, b). (iii): f: (QxQ, +,-) → (Q(√2), +,-), (x,y) →x+yv2. (iv): f: CR, a +bi+ a² + b². ...arrow_forward
- Let R be a ring. Define a functione: R[x] ×RR by the rule e(anx" +.+ax+a0,r) = anr" + +ar+ao. In other words, given the input of a polynomial f(x) € R[x] and an element r E R, e(f(x),r) is obtained by substitutingr for x in f(x). (a) Prove that, if R is a commutative ring, then e(f,r)+e(g,r) = e(f+g.r) and e(f,r)- e(8,r) = e(f8.r) for any two linear polynomials f,g € R[x] and anyrER. [In fact both facts are true for polynomials f,g of any degree. I encourage you to write up a proof of the more general versions instead, if you can. But I didn't want to burden everyone with having to use indices and arbitrarily long sums.] (b) If R is a non-commutative ring, just one of the two equations in part (a) still holds for all r, f, and g. Which one? Explain how you could make a counterexample to the other equation. [If you know a non-commutative ring, feel free to give an actual counterexample in that ring instead of "explaining how".]arrow_forwardLet I = {[xx, y = R} and J = {[²] 1z € R} E Consider the ring homomorphism : I→ R defined as (a) Show that is a ring homomorphism. (b) Use FIT for rings to show that I/JR as rings. ([*]) = x - y.arrow_forwardExpress the given linear mapping w = f(z) as a composition of a rotation R(z), magnification M(z), and a translation T(z). Then describe the action of the linear mapping in words. f(z)=(-1/2)z+1-sqrt(3)i i is not under square rootarrow_forward
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