Unformatted text preview: ,x ) be the ideal generated by 2 and x . (i) Prove that the ideal (2 ,x ) is not principal and conclude that Z [ x ] is not a principal ideal ring. (ii) Find a homomorphism f : R → Z 2 such that (2 ,x ) = Ker( f ) . 4. Prove that no two of the following rings are isomorphic: (1) R × R × R × R (with addition and multiplication given coordinate by coordinate); (2) M 2 ( R ); (3) The ring H of real quaternions. 5. Let f : R → S be a ring homomorphism. (1) Let J C S be an ideal. Prove that f1 ( J ) (equal by deﬁnition to { r ∈ R : f ( r ) ∈ J } ) is an ideal of R . (2) Prove that if f is surjective and I C R is an ideal then f ( I ) is an ideal (where f ( I ) = { f ( i ) : i ∈ I } ). (3) Show, by example, that if f is not surjective the assertion in (2) need not hold....
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 Fall '07
 Goren
 Algebra, Addition, Ring, Ring theory, Commutative ring, bi + cj

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