# nLab prefield ring

Contents

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### Constructivism, Realizability, Computability

intuitionistic mathematics

# Contents

## Idea

In the same vein that commutative rings are to integral domains and GCD rings are to GCD domains, prefield rings are to fields.

## Definition

A commutative ring $R$ is a prefield ring if the ring of fractions of $R$ is isomorphic to $R$. Equivalently, a commutative ring $R$ is a prefield ring if for all elements $a \in R$, $a$ is a cancellative element if and only if $a$ is a unit; the monoid of cancellative elements $\mathrm{Can}(F)$ is equivalent to the group of units $R^\times$.

## Examples

• The rational numbers $\mathbb{Q}$ are a prefield ring.

• A classical field $F$ is a prefield ring whose monoid of cancellative elements is the set of elements not equal to zero.

• A Heyting field $F$ is a prefield ring whose monoid of cancellative elements is the multiplicative monoid of elements apart from zero.

• The trivial ring $0$ is the unique prefield ring up to unique isomorphism such that zero is in the monoid of cancellative elements. The trivial ring is also the terminal prefield.

• Given any positive integer $n$, the integers modulo n $\mathbb{Z}/n\mathbb{Z}$ is a prefield ring whose monoid of cancellative elements consists of all integers $m$ modulo $n$ which are coprime with $n$.

• Let $F$ be a discrete field and let $\overline{F}$ be the algebraic closure of $F$. Given any non-zero polynomial $p \in \overline{F}[x]$, the quotient ring $\overline{F}[x]/p \overline{F}[x]$ is a prefield ring whose monoid of cancellative elements consists of polynomials $q \in \overline{F}[x]$ modulo $p$ such that the greatest common divisor of $p$ and $q$ is an element of the group of units $\gcd(p, q) \in \overline{F}[x]^\times$.

• Let $R$ be a unique factorization domain such that for every irreducible element $x \in R$, the ideal $x R$ is a maximal ideal. Given any non-zero element $x \in R$, the quotient ring $R/x R$ is a prefield ring whose monoid of cancellative elements consists of elements $y \in R$ modulo $x$ such that the greatest common divisor of $x$ and $y$ is an element of the group of units $\gcd(x, y) \in R^\times$.

• A prefield ring $F$ is a local ring if the set of non-cancellative elements, the zero divisors, is the Jacobson radical $J(F)$.

• Non-example: the integers $\mathbb{Z}$ are not a prefield ring.