symmetric monoidal (∞,1)-category of spectra
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.
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.
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.
Last revised on January 12, 2023 at 05:38:18. See the history of this page for a list of all contributions to it.