Let $R$ be a commutative ring with $1$ such that $R[x]$ is a PID. Show that $R$ is a field.
98540
If $F[x]$ is a principal domain does $F$ have to be necessarily a field?
23721
Prove that the Gaussian Integer's ring is a Euclidean domain
76496
Showing that every field is an integral domain.
102282
Let $R$ be a PID, and let $\pi\in R$ be prime. Is it $R/\langle\pi\rangle \times R/\langle \pi\rangle$ a cyclic $R$-module?
130466
Proving every prime ideal is maximal in a finite dimensional ring over a field.
121750
Let $R$ be a commutative, unital ring and let $I$ be an ideal of $R.$ Prove that if $I$ is maximal then $R/I$ is a field.
145879
$R$ is PID, so $R/I$ is PID, and application on $\mathbb{Z}$ and $\mathbb{N}$
23242
Commutative integral domain with d.c.c. is a field
18971
Show that a ring is commutative if it has the property that ab = ca implies b = c when $a\neq 0$
149747
Prove that if I is a prime ideal of R then I[x] is a prime ideal of R[x]
155642
Understanding divison by monic polynomial in $R[x]$ where $R$ is an arbitrary ring
55902
Show $R[x]/(x)\cong R$
46254
Why can't the Polynomial Ring be a Field?
83050
Show that $P[x]+\langle x\rangle $ is a prime ideal of $R[x]$.
19112
Ring $R= \mathbb Z_3[x]/\langle x^6-1\rangle$
34990
Ring homomorphism defined on a field
19838
Prove that in general the set of the right zero divisors is different from the set of left zero divisors in a ring R