Unformatted text preview: B ( Y ) is distinct from the one in B ( X ) . Conclude that B ( Y ) is not a subring of B ( X ) . Solution. The unit in B ( Y ) is Y . which is distinct from the unit in B ( X ) , namely, X . (iv) Prove that every element U ∈ B ( X ) satis f es U 2 = U . Solution. U ∩ U = U . 3.8 (i) If R is a domain and a ∈ R satis f es a 2 = a , prove that either a = 0 or a = 1. Solution. If a 2 = a , then 0 = a 2 − a = a ( a − 1 ) . Since R is a domain, either a = 0 or a − 1 = 0. (ii) Show that the commutative ring F ( R ) in Example 3.11(i) contains elements f ±= , 1 with f 2 = f ....
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 Fall '11
 KeithCornell
 Addition, Ring

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