Other Ring Constructions

Magma allows the construction of residue fields, localization of rings, and completion of rings. These constructions really just create appropriate rings of different categories within Magma.

Residue Class Fields

ResidueClassField(I): Rng -> Fld, Map

Given a maximal ideal \(I\) of a ring \(R\), create the residue class field \(K\) of the quotient ring \(R/I\), together with a map sending an element of \(R\) to the corresponding element of \(K\).

Localization

loc< R | a₁, ..., aᵣ >: Rng, RngElt, ..., RngElt -> Rng, Map

Given a ring \(R\) and elements \(a_1, \ldots, a_r\) of \(R\), which generate a prime ideal \(P\) of \(R\), create the localization \(L\) of \(R\) at \(P\), together with a map sending an element of \(R\) to the corresponding element of \(L\).

Localization(R, P): Rng, Rng -> Rng, Map

Given a ring \(R\) and a prime ideal \(P\) of \(R\), create the localization \(L\) of \(R\) at \(P\), together with a map sending an element of \(R\) to the corresponding element of \(L\).

Completion

comp< R | a₁, ..., aᵣ >: Rng, RngElt, ..., RngElt -> Rng, Map

Given a ring \(R\) and elements \(a_1, \ldots, a_r\) of \(R\), which generate a prime ideal or zero ideal \(P\) of \(R\), create the completion \(C\) of \(R\) at \(P\), together with a map sending an element of \(R\) to the corresponding element of \(C\).

Completion(R, P): Rng, Rng -> Rng, Map

Given a ring \(R\) and a prime ideal or zero ideal \(P\) of \(R\), create the completion \(C\) of \(R\) at \(P\), together with a map sending an element of \(R\) to the corresponding element of \(C\).

Transcendental Extension

ext< R | >: Rng -> RngUPol

Given a ring \(R\) create the univariate transcendental extension \(R[x]\) of \(R\). This is equivalent to PolynomialRing(R).

ext< R, n | >: Rng, RngIntElt -> RngMPol

Given a ring \(R\) and an integer \(n \geq 1\), create the multivariate transcendental extension \(R[x_1, \ldots, x_n]\) of \(R\). This is equivalent to PolynomialRing(R, n).