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Question III: Let M₁ and M₂ be two R-modules and let r E R. Let M₁ X M₂ = {(x,y) |x E M₁, y € M₂} with operations (x₁, y₁) + (x₂,Y₂) = (x₁+x₂, Y₁ + y₂) and r. (x, y) = (r.x, r.y) Show that M₁ X M₂ is an R-module.

Question III: Let M₁ and M₂ be two R-modules and let r E R. Let M₁ X M₂ = {(x,y) |x E M₁, y € M₂} with operations (x₁, y₁) + (x₂,Y₂) = (x₁+x₂, Y₁ + y₂) and r. (x, y) = (r.x, r.y) Show that M₁ X M₂ is an R-module.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.7: Distinguishable Permutations And Combinations
Problem 21E
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algebra module over ring
Question III: Let M₁ and M₂ be two R-modules and let r ER. Let M₁ X M₂ = operations {(x,y)|x E M₁, y € M₂} with (X₁,Y₁) + (x₂,Y₂) = (x₁+x₂, Y₁ + y₂) and r. (x, y) = (r.x, r.y) Show that M₁ X M₂ is an R-module.
Transcribed Image Text:Question III: Let M₁ and M₂ be two R-modules and let r ER. Let M₁ X M₂ = operations {(x,y)|x E M₁, y € M₂} with (X₁,Y₁) + (x₂,Y₂) = (x₁+x₂, Y₁ + y₂) and r. (x, y) = (r.x, r.y) Show that M₁ X M₂ is an R-module.
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