Show that if S is a ring and M is a freeS-module on a countably infinite set el, e2, ..then M -= M O M as Š-modules. Deducethat if R = EndS(M), then R ~= Rn as leftR-modules for all n 2 1. Hint: Show that el, e3,e5, ... and e2, e4, e6, ... span over Ssubmodules M1, M2 isomorphic to M. Showthat M = M1 O M2 as an internal direct sum.%3D

Show that if S is a ring and M is a free S-module on a countably infinite set e1, e2, .. ., then M -= M O M as Š-modules. Deduce that if R = EndS(M), then R R-modules for all n 2 1. Hint: Show that el1, e3, e5, ... and e2, e4, e6, ... span over S submodules M1, M2 isomorphic to M. Show that M = M1 O M2 as an internal direct sum. Rn as left

Show that if S is a ring and M is a free S-module on a countably infinite set e1, e2, .. ., then M -= M O M as Š-modules. Deduce that if R = EndS(M), then R R-modules for all n 2 1. Hint: Show that el1, e3, e5, ... and e2, e4, e6, ... span over S submodules M1, M2 isomorphic to M. Show that M = M1 O M2 as an internal direct sum. Rn as left

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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