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Let p: M M' be a homomorphism of R-modules and let N be an R-submodule of M. We define the image of N under o to be P(N) := {v(u) : u € N}. Show that (a) (N) is an R-submodule of M'; (b) if N = (u1,..., tum) R, then p(N) = (p(41),..,(um)) R; (c) if N admits a basis {u1,..., Um} and if p is injective, then {p(u1),...,(um)} is a basis of p(N). %3D

Let p: M M' be a homomorphism of R-modules and let N be an R-submodule of M. We define the image of N under o to be P(N) := {v(u) : u € N}. Show that (a) (N) is an R-submodule of M'; (b) if N = (u1,..., tum) R, then p(N) = (p(41),..,(um)) R; (c) if N admits a basis {u1,..., Um} and if p is injective, then {p(u1),...,(um)} is a basis of p(N). %3D

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 p: M M' be a homomorphism of R-modules and let N be an R-submodule of M. We define the image of N under y to be P(N) := {y(u) : u e N}. Show that (a) p(N) is an R-submodule of M'; (b) if N = (u1,..., Um) R, then o(N) = (p(4),...,p(um))R (c) if N admits a basis {u1,..., Um} and if p is injective, then {9(u1),...,(um)} is a basis of (N). %3D
Transcribed Image Text:Let p: M M' be a homomorphism of R-modules and let N be an R-submodule of M. We define the image of N under y to be P(N) := {y(u) : u e N}. Show that (a) p(N) is an R-submodule of M'; (b) if N = (u1,..., Um) R, then o(N) = (p(4),...,p(um))R (c) if N admits a basis {u1,..., Um} and if p is injective, then {9(u1),...,(um)} is a basis of (N). %3D
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