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Unformatted text preview: ( a ) = ( b ) for some elements a , b R if and only if a = ub for some unit u of R . Suppose a = ub for some unit u of R . Then a bR , so ( a ) ( b ) . Also if uv = 1, then b = va and we deduce that ( b ) ( a ) . Therefore ( a ) = ( b ) . Conversely suppose ( a ) = ( b ) . The result is obvious if a = 0, because then b = 0 and we can take u = 1. Therefore we may assume that a 6 = 0. Then we may write a = br and b = as for some r , s R and we have a = ars . Since a 6 = 0 and R is an integral domain, we see that 1 = rs , so r is a unit and the result follows....
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This note was uploaded on 01/02/2012 for the course MATH 4124 taught by Professor Staff during the Spring '08 term at Virginia Tech.
 Spring '08
 Staff
 Math, Algebra

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