Explanation Given information: I = M 2 ( E ) = { [ a b c d ] | a , b , c , a n d d a r e i n E } is a set of all 2 × 2 matrices over a ring E of even integers is a subset of ring M 2 ( ℤ ) . Formula Used: Definition of an ideal: A subset I of a ring R is an ideal of R provided the following conditions are satisfied: 1. I is nonempty. 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: The set of even integers is also denoted as 2 ℤ . 1) Let 2 ∈ 2 ℤ m which implies that [ 2 2 2 2 ] ∈ I Hence I is non empty. 2) Let x = [ a 1 b 1 c 1 d 1 ] , y = [ a 2 b 2 c 2 d 2 ] ∈ I where a 1 , b 1 , c 1 , c 1 , a 2 , b 2 , c 2 , d 2 ∈ 2 ℤ x + y = [ a 1 b 1 c 1 d 1 ] + [ a 2 b 2 c 2 d 2 ] = [ a 1 + a 2 b 1 + b 2 c 1 + c 2 d 1 + d 2 ] Since a 1 , b 1 , c 1 , c 1 , a 2 , b 2 , c 2 , d 2 ∈ 2 ℤ implies that a 1 + a 2 , b 1 + b 2 , c 1 + c 2 , d 1 + d 2 ∈ 2 ℤ Hence x + y = [ a 1 + a 2 b 1 + b 2 c 1 + c 2 d 1 + d 2 ] ∈ I 3) Let x = [ a 1 b 1 0 0 ] ∈ I − x = − [ a 1 b 1 0 0 ] = [ − a 1 − b 1 0 0 ] Since, a 1 , b 1 ∈ 2 ℤ implies that − a 1 , − b 1 ∈ 2 ℤ Hence − x = [ − a 1 − b 1 0 0 ] ∈ I 4) Let r = [ x 1 y 1 z 1 w 1 ] ∈ M 2 ( ℤ ) </

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ISBN: 9780357671139

Author: Gilbert

Publisher: CENGAGE L

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Chapter 6.1, Problem 26E

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To prove: set I=M2(E)={[abcd]|a,b,c, and d are in E} of all 2×2 matrices over a ring E of even integers is an ideal of the ring M2(ℤ).

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Chapter 6.1, Problem 25E

Chapter 6.1, Problem 27E

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To prove: set I = M 2 ( E ) = { [ a b c d ] | a , b , c , a n d d a r e i n E } of all 2 × 2 matrices over a ring E of even integers is an ideal of the ring M 2 ( ℤ ) .

Solution Summary: The author explains that the set of even integers is also denoted as 2mathbbZ.

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To prove: set I=M2(E)={[abcd]|a,b,c, and d are in E} of all 2×2 matrices over a ring E of even integers is an ideal of the ring M2(ℤ).

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Chapter 6 Solutions