Unformatted text preview: 98 (vii) If ϕ : R → S is a ring homomorphism and if J is an ideal in S , then the inverse image ϕ − 1 ( J ) is an ideal in R . Solution. True. (viii) If R and S are commutative rings, then the projection ( r , s ) 7→ r is a ring homomorphism R × S → R . Solution. True. (ix) If k is a fi eld and f : k → R is a surjective ring homomorphism, then R is a fi eld. Solution. True. (x) If f ( x ) = e x , then f is a unit in F ( R ) . Solution. True. 3.42 Let A be a commutative ring. Prove that a subset J of A is an ideal if and only if 0 ∈ J , u , v ∈ J implies u − v ∈ J , and u ∈ J , a ∈ A imply au ∈ J . (In order that J be an ideal, u , v ∈ J should imply u + v ∈ J instead of u − v ∈ J .) Solution. The properties of J differ from those in the de fi nition of an ideal in that (ii ’ ) u , v ∈ I implies u − v ∈ I replaces (ii) u , v ∈ I implies u + v ∈ I . Now a = − 1, says v ∈ J if and only if − v ∈ J . If (ii) holds, then u − ( − v) =...
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 Fall '11
 KeithCornell
 Ring, Abelian group, Ring theory

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