(a) Explanation Given information: The Gaussian integers modulo 3 , that is, the set S = { a + b i | a , b ∈ ℤ 3 } = { 0 , 1 , 2 , i , 1 + i , 2 + i , 2 i , 1 + 2 i , 2 + 2 i } , where we write 0 for [ 0 ] , 1 for [ 1 ] and 2 for [ 2 ] in ℤ 3 . Addition and multiplication are as in the complex numbers except that the coefficients are added and multiplied as in ℤ 3 . Thus, i 2 = − 1 as in the complex numbers and − 1 = 2 in ℤ 3 . Formula used: i) Definition of a ring: Suppose R is a set in which a relation of equality, denoted by = , and operations of addition and multiplication, denoted by + and ⋅ respectively, are defined. 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 hold in R : x ⋅ ( y + z ) = x ⋅ y + x ⋅ z and ( x + y ) ⋅ z = x ⋅ z + y ⋅ z ∀ x , y , z ∈ R ii) Definition of a commutative ring: If multiplication in ring R is commutative, then R is called commutative ring. Explanation: Let S = { a + b i | a , b ∈ ℤ 3 } = { 0 , 1 , 2 , i , 1 + i , 2 + i , 2 i , 1 + 2 i , 2 + 2 i } be the set of Gaussian integers modulo 3 with addition and multiplication as in the complex numbers except that the coefficients are added and multiplied as in ℤ 3 . First to check whether S is a ring. Addition table for S is + 0 1 2 i 1 + i 2 + i 2 i 1 + 2 i 2 + 2 i 0 0 1 2 i 1 + i 2 + i 2 i 1 + 2 i 2 + 2 i 1 1 2 0 1 + i 2 + i i 1 + 2 i 2 + 2 i 2 i 2 2 0 1 2 + i i 1 + i 2 + 2 i 2 i 1 + 2 i i i 1 + i 2 + i 2 i 1 + 2 i 2 + 2 i 0 1 2 1 + i 1 + i 2 + i i 1 + 2 i 2 + 2 i 2 i 1 2 0 2 + i 2 + i i 1 + i 2 + 2 i 2 i 1 + 2 i 2 0 1 2 i 2 i 1 + 2 i 2 + 2 i 0 1 2 i 1 + i 2 + i 1 + 2 i 1 + 2 i 2 + 2 i 2 i 1 2 0 1 + i 2 + i i 2 + 2 i 2 + 2 i 2 i 1 + 2 i 2 0 1 2 + i i 1 + i Multiplication table for S is ⋅ 0 1 2 i 1 + i 2 + i 2 i 1 + 2 i 2 + 2 i 0 0 0 0 0 0 0 0 0 0 1 0 1 2 i 1 + i 2 + i 2 i 1 + 2 i 2 + 2 i 2 0 2 1 2 i 2 + 2 i 1 + 2 i i 2 + i 1 + i i 0 i 2 i 2 2 + i 2 + 2 i 1 1 + i 1 + 2 i 1 + i 0 1 + i 2 + 2 i 2 + i 2 i 1 1 + 2 i 2 i 2 + i 0 2 + i 1 + 2 i 2 + 2 i 1 i 1 + i 2 i 2 2 i 0 2 i i 1 1 + 2 i 1 + i 2 2 + 2 i 2 + i 1 + 2 i 0 1 + 2 i 2 + i 1 + i 2 2 i 2 + 2 i i 1 2 + 2 i 0 2 + 2 i 1 + i 1 + 2 i i 2 2 + i 1 2 i From the table, First condition: From the above addition table, Let x = m 1 + n 1 i ∈ S and y = m 2 + n 2 i ∈ S where m 1 , m 2 , n 1 , n 2 ∈ ℤ 3 x + y = ( m 1 + n 1 i ) + ( m 2 + n 2 i ) = ( m 1 + m 2 ) + ( n 1 + n 2 ) i = m + n i ∈ S Where, m 1 + m 2 = m ∈ ℤ 3 and n 1 + n 2 = n ∈ ℤ 3 Therefore, S is closed under addition (b) To determine Whether the set S has unity. (c) To determine Whether the set S is an integral domain. (d) To determine Whether the set S is a field.

8th Edition

ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Question

Chapter 5.2, Problem 12E

(a)

To determine

Whether S is a commutative ring.

(b)

To determine

Whether the set S has unity.

(c)

To determine

Whether the set S is an integral domain.

(d)

To determine

Whether the set S is a field.

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Chapter 5.2, Problem 11E

Chapter 5.2, Problem 13E

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Chapter 5 Solutions

Elements Of Modern Algebra

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Whether S is a commutative ring.

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

Whether S is a commutative ring.

Whether the set S has unity.

Whether the set S is an integral domain.

Whether the set S is a field.

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Chapter 5 Solutions