Explanation Given information: Ring W and ring S are isomorphic rings. Formula used: 1) Definition of ring isomorphism: Let R and R ' denote two rings. A mapping ϕ : R → R ' is a ring isomorphism from R to R ' provided the following conditions hold: 1. ϕ is one-to-one correspondence (a bijection) from R to R ' . 2. ϕ ( x + y ) = ϕ ( x ) + ϕ ( y ) for all x and y in R . 3. ϕ ( x ⋅ y ) = ϕ ( x ) ⋅ ϕ ( y ) for all x and y in R . 2) The multiplication of ordered pairs is given by ( x , y ) ⋅ ( z , w ) = ( x z − y w , x w + y z ) 3) The addition of ordered pairs is given by ( x , y ) + ( z , w ) = ( x + z , y + w ) Proof: Define a mapping ϕ : W → S such that ϕ ( ( x , y ) ) = [ x − y y x ] . A mapping ϕ : W → S is an isomorphism from W to S if it satisfies the following conditions: 1. ϕ is one-to-one correspondence (a bijection) from W to S : Let ϕ ( ( x 1 , y 1 ) ) = ϕ ( ( x 2 , y 2 ) ) By using mapping ϕ , [ x 1 − y 1 y 1 x 1 ] = [ x 2 − y 2 y 2 x 2 ] Corresponding elements of equal matrices are also equal. ⇒ x 1 = x 2 , y 1 = y 2 ⇒ ( x 1 , y 1 ) = ( x 2 , y 2 ) Therefore, ϕ is one-to-one. Let [ x − y y x ] ∈ S ; then there exists ( x , y ) ∈ W such that ϕ ( ( x , y ) ) = [ x − y y x ] . Therefore, ϕ is onto. 2. ϕ ( x + y ) = ϕ ( x ) + ϕ ( y ) for all x and y in W : Let ( x 1 , y 1 ) , ( x 2 , y 2 ) ∈ W

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Author: Gilbert

Publisher: YUZU

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Chapter 5.3, Problem 9E

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To prove: The mapping from W to S is an isomorphism if the ring W and ring S are

isomorphic rings.

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Chapter 5.3, Problem 8E

Chapter 5.3, Problem 10E

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To prove: The mapping from W to S is an isomorphism if the ring W and ring S are isomorphic rings.

Solution Summary: The author explains that the mapping from W to S is an isomorphism, provided the following conditions hold.

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To prove: The mapping from W to S is an isomorphism if the ring W and ring S are isomorphic rings.

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Chapter 5 Solutions

To prove: The mapping from W to S is an isomorphism if the ring W and ring S are

isomorphic rings.