# nov08 - Math 3124 Tuesday, November 8 November 8, Ungraded...

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Math 3124 Tuesday, November 8 November 8, Ungraded Homework Problem 24.5 on page 124. Prove that Z [ 2 ] (that is { a + b 2 | a , b Z } ) is a ring. We have the two operations of addition and multiplication as usual, so associativity and distributivity are OK. The only thing that needs to be checked is that we have closure under addition, subtraction (inverse for addition) and multiplication. Let a + b 2, c + d 2 Z [ 2 ] (equivalently, a , b , c , d Z ). ( a + b 2 )+( c + d 2 ) = ( a + c )+( b + d ) 2 Z [ 2 ] because ( a + c ) , ( b + d ) Z . - ( a + b 2 ) = - a - b 2 Z [ 2 ] because - a , - b Z . ( a + b 2 )( c + d 2 ) = ( ac + 2 bd )+( ad + bc ) 2 Z [ 2 ] because ( ac + 2 bd ) , ( ad + bc ) Z . Problem 24.15 on page 125. Let E denote the set of even integers. Prove that with the usual addition, and with multiplication deﬁned by m * n = mn / 2, then E is a ring. Does it have a unity? Addition makes

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## This note was uploaded on 01/02/2012 for the course MATH 3124 taught by Professor Parry during the Fall '08 term at Virginia Tech.

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nov08 - Math 3124 Tuesday, November 8 November 8, Ungraded...

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