Explanation Given information: b , a 1 , a 2 , a 3 , . . . a n are the elements of a ring R . Formula used: Definition of a ring: Suppose R is a set in which a relation of equality is denoted by = , and the operations of addition and multiplication are denoted by + and ⋅ respectively, then R is a ring with respect to these operations if the following conditions are satisfied: 1) R is closed under addition: x ∈ R , y ∈ R ⇒ x + y ∈ R 2) Addition in R is associative: ( x + y ) + z = x + ( y + z ) ∀ x , y , z ∈ R 3) R contains an additive identity 0 : x + 0 = 0 + x = x ∀ x ∈ R 4) R contains an additive inverse: for each x in R , there exists − x in R such that x + ( − x ) = ( − x ) + x = 0 . 5) Addition in R is commutative: x + y = y + x ∀ x , y ∈ R 6) R is closed under multiplication: x ∈ R , y ∈ R ⇒ x ⋅ y ∈ R 7) Multiplication in R is associative: ( x ⋅ y ) ⋅ z = x ⋅ ( y ⋅ z ) ∀ x , y , z ∈ R 8) Two distributive laws holds in R : x ⋅ ( y + z ) = x ⋅ y + x ⋅ z and ( x + y ) ⋅ z = x ⋅ z + y ⋅ z ∀ x , y , z ∈ R Explanation: For n ≥ 2 , let P n = b ( a 1 + a 2 + a 3 + . . . + a n ) = b a 1 + b a 2 + b a 3 + . . . + b a n . By using mathematical induction, For n = 2 , P 2 = b ( a 1 + a 2 ) = b a 1 + b a 2 This is true by left distributive law of ring R

8th Edition

ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Chapter 5.1, Problem 47E

To determine

To prove:

Let n ≥ 2 be a positive integer, and b, a1, a2, a3, . . . an denote elements of a ring R.Then, we have

a) b(a1+a2+ a3+ . . . +an)=ba1+ba2+b a3+ . . . +ban

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Chapter 5.1, Problem 46E

Chapter 5.1, Problem 48E

Chapter 5 Solutions

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To prove: Let n ≥ 2 be a positive integer, and b , a 1 , a 2 , a 3 , . . . a n denote elements of a ring R .Then, we have a) b ( a 1 + a 2 + a 3 + . . . + a n ) = b a 1 + b a 2 + b a 3 + . . . + b a n

Solution Summary: The author explains that R is a ring if the following conditions are satisfied.

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To prove:

Let n ≥ 2 be a positive integer, and b, a1, a2, a3, . . . an denote elements of a ring R.Then, we have

a) b(a1+a2+ a3+ . . . +an)=ba1+ba2+b a3+ . . . +ban

Want to see the full answer?

Chapter 5 Solutions

To prove: Let n ≥ 2 be a positive integer, and b , a 1 , a 2 , a 3 , . . . a n denote elements of a ring R .Then, we have a) b ( a 1 + a 2 + a 3 + . . . + a n ) = b a 1 + b a 2 + b a 3 + . . . + b a n

Solution Summary: The author explains that R is a ring if the following conditions are satisfied.

Videos

To prove: Let n ≥ 2 be a positive integer, and b, a1, a2, a3, . . . an denote elements of a ring R.Then, we have a) b(a1+a2+ a3+ . . . +an)=ba1+ba2+b a3+ . . . +ban

Want to see the full answer?

Chapter 5 Solutions

To prove:

Let n ≥ 2 be a positive integer, and b, a1, a2, a3, . . . an denote elements of a ring R.Then, we have

a) b(a1+a2+ a3+ . . . +an)=ba1+ba2+b a3+ . . . +ban