If $R$ is a ring (Integral Domain) then $R[x_1,x_2]$ is not a Principal Ideal Domain. Is it a unique factorization domain? Any hints on how to prove its not a Principal Ideal Domain?
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Hint: Let $R$ have more than one element. To show $R[x_1,x_2]$ is not a Principal Ideal Domain, consider the polynomials with $0$ constant term. There are many other non-principal ideals.