Unformatted text preview: x zero) of k [ x ] . Prove that x 2 is irreducible but not prime in R . Suppose x 2 = fg where f , g ∈ R . Since deg ( fg ) = deg ( f )+ deg ( g ) and R has no element of degree 1, we see that one of f , g has degree zero, say deg ( f ) = 0. But this implies 0 6 = f ∈ k and hence f is a unit. This shows that x 2 is irreducible. However x 2 is not prime because x 2 ± ± x 3 x 3 , but x 2x 3 ....
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 Fall '07
 PALinnell
 Math, Algebra, Ring theory, nonzero ring homomorphism, nonzero zerodivisors

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