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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 Spring '08
 Staff
 Math, Algebra, Integral domain, Ring theory, Commutative ring, Principal ideal domain

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