Explanation Formula used: Definition of ideal: A subset I of a ring R is an ideal of R provide the following conditions are satisfied: 1. I is non-empty. 2. x ∈ I and y ∈ I implies x + y ∈ I . 3. x ∈ I implies − x ∈ I . 4. x ∈ I and r ∈ R implies x r and r x are in I . Proof: Let S is the set of all ordered pairs of the form ( [ m ] , x ) where [ m ] ∈ ℤ n and x ∈ R . Equality in S is defined by, ( [ m ] , x ) = ( [ k ] , y ) if and only if [ m ] = [ k ] and x = y Addition and multiplication in S is defined by, ( [ m ] , x ) + ( [ k ] , y ) = ( [ m + k ] , x + y ) ( [ m ] , x ) ⋅ ( [ k ] , y ) = ( [ m k ] , m y + k x + x y ) R ' = { ( [ 0 ] , x ) | x ∈ R } 1. As 0 ∈ R , then ( [ 0 ] , 0 ) ∈ R ' Therefore, R ' is non-empty. 2. Let ( [ 0 ] , x ) ∈ R ' and ( [ 0 ] , y ) ∈ R ' By using definition of operation in S , ( [ 0 ] , x ) + ( [ 0 ] , y ) = ( [ 0 ] , x + y ) By using closure property of addition in ring R , As x , y ∈ R ⇒ x + y ∈ R Thus, ( [ 0 ] , x + y ) ∈ R ' Therefore, ( [ 0 ] , x ) + ( [ 0 ] , y ) ∈ R ' 3. Let ( [ 0 ] , x ) ∈ R ' with x ∈ R By using additive inverse property in ring R , x ∈ R ⇒ − x ∈ R Thus, − ( [ 0 ] , x ) = ( [ 0 ] , − x ) ∈ R ' Therefore, ( [ 0 ] , x ) ∈ R ' . That is − ( [ 0 ] , x ) ∈ R '

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ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Chapter 6.3, Problem 25E

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To prove: The set R'={([0],x) | x∈R} is an ideal of S={([m],x) | [m]∈ℤn,x∈R}

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Chapter 6.3, Problem 24E

Chapter 6.3, Problem 26E

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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

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To prove: The set R ' = { ( [ 0 ] , x ) | x ∈ R } is an ideal of S = { ( [ m ] , x ) | [ m ] ∈ ℤ n , x ∈ R }

Solution Summary: The author explains how the set R' is an ideal of S provided the following conditions are satisfied.

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To prove: The set R'={([0],x) | x∈R} is an ideal of S={([m],x) | [m]∈ℤn,x∈R}

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Chapter 6 Solutions