3- For any R-module ,let M [x]denote the set of all polynomials in x with coefficient in M, that is to say expression of the form m.+mix+...+m,x.(mi eM). Defining the product of an element of R[x] and an element of M[x] in the obvious way, show that M[x] is an R[x]-module. Show that M[x] = R[x] ®R M.
3- For any R-module ,let M [x]denote the set of all polynomials in x with coefficient in M, that is to say expression of the form m.+mix+...+m,x.(mi eM). Defining the product of an element of R[x] and an element of M[x] in the obvious way, show that M[x] is an R[x]-module. Show that M[x] = R[x] ®R M.
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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![3- For any R-module ,let M [x]denote the set of all polynomials in x with coefficient in M, that is to
say expression of the form m.+mix+...+m,x.(mi eM). Defining the product of an element of R[x]
and an element of M[x] in the obvious way, show that M[x] is an R[x]-module. Show that M[x] =
R[x] ®R M.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feff78ae8-57e1-4369-9056-f925480f941c%2F102138fd-c196-4eb0-8404-19a85dc07870%2Fj6mdmh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3- For any R-module ,let M [x]denote the set of all polynomials in x with coefficient in M, that is to
say expression of the form m.+mix+...+m,x.(mi eM). Defining the product of an element of R[x]
and an element of M[x] in the obvious way, show that M[x] is an R[x]-module. Show that M[x] =
R[x] ®R M.
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