Suppose that is a surjective ring homomorphism. Suppose that is an irreducible element. Is it true that is also irreducible? Prove it or give a counter-example.
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R = Z[x], r = x + 4, S = Z, φ:R-->S given by φ(anx^n +.....+a1x + a0) = a0. x+4 is irreducible in Z[x], but 4 is not irreducible in Z.
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