nLab prefield ring

Contents

Context

Algebra

Constructivism, Realizability, Computability

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

Examples

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

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

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

  • The trivial ring 00 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 nn, the integers modulo n /n\mathbb{Z}/n\mathbb{Z} is a prefield ring whose monoid of cancellative elements consists of all integers mm modulo nn which are coprime with nn.

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

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

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

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

See also

Last revised on January 23, 2023 at 18:02:07. See the history of this page for a list of all contributions to it.