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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.tmix+...+m,x .(mi eM). Defining the product of an element ofR[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.tmix+...+m,x .(mi eM). Defining the product of an element ofR[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
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 mo+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] OR M.
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 mo+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] OR M.
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