Let R be a ring and M a module over R. Let S be a non- empty set and let {x; : i € S} be a basisof M. Let N be an R-module and let {y; : i € S} be a family of elements of N. Then there existsa unique homomorphism f: M→N such thatf(x₁) = y₁ Vie S.

Given that R be a ring and M a module over R. Let S be a non-empty set and let {xi : i∈ S } be a…

Answered: Let R be a ring and M a module over R.…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.2: Ring Homomorphisms

Problem 14E: 14. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.

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Let R be a ring and M a module over R. Let S be a non- empty set and let {x; : i E S} be a basis of M. Let N be an R-module and let {y;:i € S} be a family of elements of N. Then there exists a unique homomorphism f:M→N such that f(xi) = y₁ Vies.