The following generic ring functions are applicable to the ring of integers and its elements.

Subsections

• Related Structures
• Numerical Invariants
• Ring Predicates and Booleans

Category(Z) : RngInt -> Cat

Parent(Z) : RngInt -> PowerStructure

PrimeRing(Z) : RngInt -> RngInt

Center(Z) : RngInt -> RngInt

Create the abelian group of integers under addition. This returns an infinite (additive) abelian group A (of rank 1) together with a map from the ring of integers Z to A, sending 1 to 1.

UnitGroup(Z) : RngInt -> GrpAb, Map

Create the abelian group of invertible integers, that is, an abelian group isomorphic to the multiplicative subgroup < - 1 >. This returns an (additive) abelian group A of order 2 together with a map from the ring of integers Z to A, sending -1 to 1.

Create the field of fractions Q of the ring of rational integers.

Given Z, the ring of integers or an ideal of it, and an element n of Z, create the ideal aZintersect Z of the ring of integers. Note that this creates an ideal, not just a subring.

Characteristic(Z) : RngInt -> RngIntElt

IsCommutative(Z) : RngInt -> BoolElt

IsUnitary(Z) : RngInt -> BoolElt

IsFinite(Z) : RngInt -> BoolElt

IsOrdered(Z) : RngInt -> BoolElt

IsField(Z) : RngInt -> BoolElt

IsEuclideanDomain(Z) : RngInt -> BoolElt

IsPID(Z) : RngInt -> BoolElt

IsUFD(Z) : RngInt -> BoolElt

IsDivisionRing(Z) : RngInt -> BoolElt

IsEuclideanRing(Z) : RngInt -> BoolElt

IsPrincipalIdealRing(Z) : RngInt -> BoolElt

IsDomain(Z) : RngInt -> BoolElt

Z eq R : RngInt, Rng -> BoolElt

Z ne R : RngInt, Rng -> BoolElt