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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online