Prime ideals in a quotient of a DVR

Question

Suppose $R$ is a DVR. So $R$ has two prime ideals - $(0)$ and $(p)$ ($p$ the uniformizer of the maximal ideal). All other ideals in $R$ are powers of $(p)$, i.e. of the form $(p^k), k\geq 2$. I'm interested in the prime ideals of the quotient $R/(p^m)$ for some fixed $m\geq 2$. I know that the (prime) ideals of the quotient are the (prime) ideals of $R$ that contain $(p^m)$, so, the only prime ideal in $R/(p^m)$ is $(p)$? Isn't that weird? I thought fields were the only rings with one prime ideal.

Answer 1

Yes, the only prime ideal of $R/(p^m)$ is $\mathfrak m=(p)/(p^m)$.
The consideration of local rings $(A,\mathfrak m)$ with only one prime ideal $\mathfrak m$ but with plenty of nilpotent ideals (the elements of that maximal ideal $\mathfrak m$) has proved essential in Grothendieck's revolutionary vision of algebraic geometry.
For example the beginning of the theory of the deformation of a scheme $X$ is the study of first order infinitesimal deformations of $X$ over $S=\operatorname {Spec}A= \operatorname {Spec}\frac { k[[t]]}{ (t^2)}= \operatorname {Spec}\frac { k[t]}{ (t^2)}$, namely flat morphisms $\pi:\mathcal X\to S$ whose fibre over $\operatorname {Spec}k\subset S$ is $X$.

Edit: bibliography
At the OP's request I'll add two references.
a) Hartshorne's Deformation theory is the definitive treatmetnt , as one could expect from that remarkable author.
A preliminary manuscript can be freely downloaded here. As far as I can judge the book consists of just that manuscript, but with exercises added.
b) Osserman's handout is a pleasant, user-friendly introduction to the subject.
He proves for example that for a smooth variety over a field $k$ the first-infinitesimal deformations of $X$ are parametrized by $H^1(X,T_X)$.
As an Exercise I suggest proving, using the Euler sequence for example, that this $k$- vector space is zero for $X=\mathbb P^n$.
Another exercise is to prove that for a smooth projective curve $C$ of genus $g\geq 2$ the vector space $H^1(C,T_C)$ has dimension $3g-3$.