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Let R be a commutative unitary ring and let M be an R-module. For a fixed reR let rM = {rx : x€M} and My = {x€M : rx = = 0}. a. Show that rM and M₁ are submodules of M. b. Let R = Z and M = Z/nZ. Suppose n = rs where gcd(r,s) = 1 (that is 1). Show that rM = Ms. a, bez such that ra + sb : =

Let R be a commutative unitary ring and let M be an R-module. For a fixed reR let rM = {rx : x€M} and My = {x€M : rx = = 0}. a. Show that rM and M₁ are submodules of M. b. Let R = Z and M = Z/nZ. Suppose n = rs where gcd(r,s) = 1 (that is 1). Show that rM = Ms. a, bez such that ra + sb : =

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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Let R be a commutative unitary ring and let M be an R-module. For a fixed reR let rM = {rx : xeM} and M, = {xeM : rx = 0}. a. Show that rM and Mr are submodules of M. b. Let R = Z and M = Z/nZ. Suppose n = rs where gcd (r,s) = 1 (that is Ba, bez such that ra + sb = 1). Show that rM = Ms.
Transcribed Image Text:Let R be a commutative unitary ring and let M be an R-module. For a fixed reR let rM = {rx : xeM} and M, = {xeM : rx = 0}. a. Show that rM and Mr are submodules of M. b. Let R = Z and M = Z/nZ. Suppose n = rs where gcd (r,s) = 1 (that is Ba, bez such that ra + sb = 1). Show that rM = Ms.
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