Direct Limit of Polynomials

Question

This is probably a very stupid question, but what is the direct limit $\lim k[x]/x^n$, where the map $k[x]/x^n \to k[x]/x^{n+1}$ is given by multiplication by $x$. For $\mathbb{Z}/p$ this is the Prufer group. I know I can write it as an infinite disjoint union quotienting a relation, but I am looking for an explicit description.

Answer 1

If $R$ is any commutative ring and $p \in R$ is an element, then you can consider the $R$-module $$R/p^{\infty} := \varinjlim R/p^n$$ with the transition maps $p : R/p^n \to R/p^{n+1}$. Many properties which work for $R=\mathbb{Z}$ and prime numbers $p$ (Prüfer group) carry over to the case that $ R$ is a PID and $p$ is a prime element, as in your example. I do not think that the restriction to polynomial rings is beneficial here to understand what is going on. For example:

• $R/p^{\infty}$ is isomorphic to the submodule $R[1/p]/R$ of $Q(R)/R$.
• $R/p^{\infty}$ is a torsion divisible $R$-module. (Actually, one can show that every divisible $R$-module is a direct sum of copies of $R/p^{\infty}$ and $Q(R)$.)
• $R/p^{\infty}$ admits a presentation $\langle x_1,x_2,\dotsc : p x_1 = 0, \, x_1 = p x_2, \, x_2 = p x_3,\, \dotsc \rangle$ as $R$-module.
• There is an isomorphism of rings $\mathrm{End}(R/p^{\infty}) \cong \varprojlim R/p^n$.