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Unformatted text preview: 98 (vii) If : R S is a ring homomorphism and if J is an ideal in S , then the inverse image 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