Let R = R[x] be the ring of real polynomials in the variable x, and let I = be the ideal of R generated by 2². For an integer d≥ 1, we denote by Rd the set of polynomials in R of degree less or equal than d. Show that the set V = {(f, g) € R₁ × Rd : f g € I} is an R-module (i.e., V is a real vector space), where r(f, g) = (rf, rg), for r ER and (f, g) € V.
Let R = R[x] be the ring of real polynomials in the variable x, and let I = be the ideal of R generated by 2². For an integer d≥ 1, we denote by Rd the set of polynomials in R of degree less or equal than d. Show that the set V = {(f, g) € R₁ × Rd : f g € I} is an R-module (i.e., V is a real vector space), where r(f, g) = (rf, rg), for r ER and (f, g) € V.
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 =
R[x] be the ring of real polynomials in the variable x, and let
I = <r²> be the ideal of R generated by 2². For an integer d≥ 1, we denote
by Rd the set of polynomials in R of degree less or equal than d. Show that the
set V = {(f, g) € Rd x Rd : f g € I} is an R-module (i.e., V is a real vector
space), where r(f, g) = (rf, rg), for r ER and (f, g) EV.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb04e43c2-1385-4673-b483-013357b8a1b6%2F8e8d7654-73d7-4a2e-894a-52a7214b66c6%2Fq0gd61f_processed.png&w=3840&q=75)
Transcribed Image Text:Let R =
R[x] be the ring of real polynomials in the variable x, and let
I = <r²> be the ideal of R generated by 2². For an integer d≥ 1, we denote
by Rd the set of polynomials in R of degree less or equal than d. Show that the
set V = {(f, g) € Rd x Rd : f g € I} is an R-module (i.e., V is a real vector
space), where r(f, g) = (rf, rg), for r ER and (f, g) EV.
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