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Integral domains, Principle Ideal domains, Fields

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For each of the following rings answer the following questions:
1. Is it an integral domain?
2. Is it a principal ideal domain (PID)?
3. Is it a field?
Give reasons (i.e. short proofs, if needed) for your answer.
1. Z/13Z;
2. Z/20Z;
3. ZÃ?Z with componentwise addition and multiplication;
4. Q[X]/(f) with f=X^2+X+1;
5. C[X]/(f) with f=X^2+X+1;
6. Z[X];
7. R[X] with R=Z/4Z;
8. R[X]/I with R=Z/2Z and I=(X^2+X+1).

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Solution Summary

This solution evaluates integral domains, principle ideal domains, and fields.