Properties of $R/I$

Question

Let $R$ be an integral domain, and let $a$ be an irreducible element of $R$. Let $I$ be the ideal of $R$ generated by $a$.

1.If $R$ is a principal ideal domain, $R/I$ is a field ?

True. Since $a$ is irreducible, $I$ is maximal among proper principal ideals of $R.$ Since $R$ is PID, every ideal of $R$ is the form of $(r),$ $r \in R.$ Hence, $I$ is maximal and $R/I$ is a field.

2.If $R$ is a unique factorization domain, then $R/I$ is a field ?

False ?

Could someone advise me on the correct approach to this problem?

Thank you.

Answer 1

As before, recall that $R/I$ is a field iff $I$ maximal. Consider $\mathbb{Z}[X]$, the element $2$; and the ideals $(2)$, $(2,X)$.

Answer 2

2) is false in general. For example, consider polynomials in 2 variables over a field $\mathbb{k}[x,y]$ which is a UFD but not a PID. Then $y$ is irreducible but $\mathbb{k}[x,y]/(y) = \mathbb{k}[x]$