nLab prime power local ring

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Context

Algebra

higher algebra

universal algebra

Contents

Idea

Prime power local rings are the integers modulo $n$ which are local rings, in the same way that prime fields are the integers modulo $n$ which are fields.

Definition

For $p \geq 2$ a prime number and for $n \geq 1$ a positive natural number, the quotient $\mathbb{Z}/p^n\mathbb{Z}$ of the ring of integers by the prime power $p^n$ is a local ring, a prime power local ring.

Properties

The ideal of non-invertible elements is the ideal $p(\mathbb{Z}/p^n\mathbb{Z})$, and the quotient of $\mathbb{Z}/p^n\mathbb{Z}$ by the ideal of non-invertible elements is the prime field $\mathbb{Z}/p\mathbb{Z}$. Since the ideal of non-invertible elements is a nilradical, every prime power local ring is an Artinian ring, which implies that it is a local Artinian ring.

The underlying abelian group of a prime power local ring is a p-primary group.

The cardinality of the group of units of $\mathbb{Z}/p^n\mathbb{Z}$ (the Euler totient function $\phi(p^n)$) is given by $\phi(p^n) = p^{n - 1} (p - 1)$.

In addition, every prime power local ring with trivial nilradical is a prime field.