nLab prefield ring

Idea

A commutative ring in which all cancellative elements are invertible.

“Prefield ring” is a placeholder name for a concept which may or may not have another name in the mathematics literature. The idea however is that prefield rings are to fields as commutative rings are to integral domains and GCD rings are to GCD domains.

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.

See also

• commutative ring

• GCD ring

• Bézout ring

• unique factorization ring

• Euclidean ring

• ring of fractions

• prevector space

• field