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 = ( a 1 + a 2 + a 3 + . . . + a n ) b = a 1 b + a 2 b + a 3 b + . . . + a n b By using mathematical induction, For n = 2 , P 2 = ( a 1 + a 2 ) b = a 1 b + a 2 b This is true by right distributive law in ring R

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ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Chapter 5.1, Problem 48E

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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, b) (a1+a2+ a3+ . . . +an)b=a1b+a2b+a3b+ . . . +anb

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Chapter 5.1, Problem 47E

Chapter 5.1, Problem 49E

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Assume {u1, U2, u3, u4} does not span R³. Select the best statement. A. {u1, U2, u3} spans R³ if u̸4 is a linear combination of other vectors in the set. B. We do not have sufficient information to determine whether {u₁, u2, u3} spans R³. C. {U1, U2, u3} spans R³ if u̸4 is a scalar multiple of another vector in the set. D. {u1, U2, u3} cannot span R³. E. {U1, U2, u3} spans R³ if u̸4 is the zero vector. F. none of the above

Select the best statement. A. If a set of vectors includes the zero vector 0, then the set of vectors can span R^ as long as the other vectors are distinct. n B. If a set of vectors includes the zero vector 0, then the set of vectors spans R precisely when the set with 0 excluded spans Rª. ○ C. If a set of vectors includes the zero vector 0, then the set of vectors can span Rn as long as it contains n vectors. ○ D. If a set of vectors includes the zero vector 0, then there is no reasonable way to determine if the set of vectors spans Rn. E. If a set of vectors includes the zero vector 0, then the set of vectors cannot span Rn. F. none of the above

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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, b) ( a 1 + a 2 + a 3 + . . . + a n ) b = a 1 b + a 2 b + a 3 b + . . . + a n b

Solution Summary: The author explains that R is a ring with respect to the operations of addition and multiplication 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, b) (a1+a2+ a3+ . . . +an)b=a1b+a2b+a3b+ . . . +anb

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Chapter 5 Solutions