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".]
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".]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![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".]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F75695e3c-64c6-4e91-aede-51c9bf48cab1%2Fe0e0353d-7ddf-4ad6-a76f-c3231dccd62d%2F1n955cv_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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".]
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