The given ring is .
Using this ring ,we have to show
1.The quotient ring of a UFD need not be UFD.
2.An irreducible elements of a UFD need not be a prime.
As we Know that , is a Euclidean domain.
Again, every Euclidean domain is PID and every PID is UFD.
Hence, is a UFD.
Now consider the ideal ,
.
Then ,
clearly,
but, .
Hence, is not a integral domain.
Therefore, it is not a unique factorization domain.
Since, To become a UFD ,It must be a integral domain .
2. As is a PID ,so every irreducible elements is a prime.
Hence, the question is wrong in this case.
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