nLab zero-dimensional ring

Defintion

A zero-dimensional ring is a commutative ring $R$ such that for all elements $x \in R$ there exists an element $a \in R$ and a natural number $n \in \mathbb{N}$ such that $x^n = a x^{n + 1}$

Properties

Every zero-dimensional ring is a “prefield ring” in that every cancellative element is invertible.

Examples

• Every Boolean ring is a zero-dimensional ring.

• Every discrete field is a zero-dimensional ring which is a local ring and a reduced ring.

• Every commutative von Neumann regular ring is a zero-dimensional ring which is a reduced ring.

• Every local Artinian ring is a zero-dimensional ring which is also a local ring. In constructive mathematics, only the residually discrete local Artinian rings are zero-dimensional local rings.

• Every Artinian ring is a zero-dimensional ring which is also a Noetherian ring and a coherent ring.

BHK interpretation

The BHK interpretation of a zero-dimensional ring states that for all elements $x \in R$ one can construct an element $a \in R$ and a natural number $n \in \mathbb{N}$ such that $x^n = a x^{n + 1}$. This is equivalent to adding to the ring the structure of an endofunction $\alpha:R \to R$ and a function to the natural numbers $\nu:R \to \mathbb{N}$, where the equational axiom is then for all elements $x \in R$, $x^{\nu(x)} = \alpha(x) x^{\nu(x) + 1}$.

It is unknown if it is possible to construct the above structure from the usual notion of a zero-dimensional ring. However, given a ring $R$ with an endofunction $\alpha:R \to R$, if for all elements $x \in R$ and natural numbers $n:\mathbb{N}$, the equality $x^n = \alpha(x) x^{n + 1}$ is decidable (such as the case in classical mathematics), then the existential quantifier

$$\exists n \in \mathbb{N}.x^n = \alpha(x) x^{n + 1}$$

has a choice operator, from which one can construct the function $\nu:R \to \mathbb{N}$ such that for all elements $x \in R$, $x^{\nu(x)} = \alpha(x) x^{\nu(x) + 1}$.

Related concepts

• Artinian ring

• von Neumann regular ring

References

• Henri Lombardi, Claude Quitté (2010): Commutative algebra: Constructive methods (Finite projective modules) Translated by Tania K. Roblo, Springer (2015) [doi:10.1007/978-94-017-9944-7, pdf]