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Unformatted text preview: nition of ideal, or note that R { } is the kernel of the homomorphism R S S given by ( r , s ) 7 s . (iii) Show that R S is not a domain if neither R nor S is the zero ring. Solution. ( 1 , ) ( , 1 ) = ( , ) . 3.54 (i) If R and S are commutative rings, prove that U ( R S ) = U ( R ) U ( S ), where U ( R ) is the group of units of R . Solution. We f rst prove that ( r , s ) is a unit in R S if and only if r is a unit in R and s is a unit in S . If r , s ) is a unit in R S , then there is ( a , b ) R S with ( r , s )( a , b ) = ( 1 , 1 ) . It follows that ra = 1 and sb = 1; that is, r is a unit in R and s is a unit in S ....
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 Fall '11
 KeithCornell

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