[Next][Prev] [Right] [Left] [Up] [Index] [Root]

Ideals of Z

The theory of ideals of Z is very elementary but for completeness the general machinery for ring ideals applies. Such ideals will have type RngInt, that is, the same type as the ring of integers itself (ideal<Integers() | 1>).

In the case of Z any subring is an ideal so that the sub-constructor creates the same object as does the ideal-constructor.

ideal< R | a > : RngInt, RngIntElt -> RngIntRes
Given the ring of integers Z and an integer a, return the ideal of Z generated by a.

Example RngIntRes_residue-ring (H19E1)

We construct some ideals of Z.

> Z := IntegerRing();
> I13 := ideal< Z | 13 >;
> I13;
Ideal of Integer Ring generated by 13
> 1 in I13;
false
> 0 in I13;
true
> -13 in I13;
true
> I0 := ideal< Z | 0 >;
> 0 in I0;
true
> 1 in I0;
false

We check that that Z is regarded as an ideal.

> I1 := ideal< Z | 1 >;
> I1 eq Z;
true

 [Next][Prev] [Right] [Left] [Up] [Index] [Root]

Version: V2.19 of Mon Dec 17 14:40:36 EST 2012