Question: Consider the ring Z[x] and let J = x + 1 . Explain why J is prime. Show that J is not maximal. Conclude that

Consider the ring Z[x] and let J = x + 1 .

Explain why J is prime.

Show that J is not maximal.

Conclude that Z [x]/ J is an integral domain but not a field.

Observe that J is the kernel of the evaluation homomorphism 1:Z[x]Z

Consequently, Z[x]/JZ by the first isomorphism theorem for rings.

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