53.3 Rings With One

A ring-with-one in GAP is a ring (see IsRing) that is also a magma-with-one (see IsMagmaWithOne).

Note that the identity and the zero of a ring-with-one need not be distinct. This means that a ring that consists only of its zero element can be regarded as a ring-with-one.

This is especially useful in the case of finitely presented rings, in the sense that each factor of a ring-with-one is again a ring-with-one.

In the first form RingWithOne returns the smallest ring with one that contains all the elements r, s... etc. In the second form RingWithOne returns the smallest ring with one that contains all the elements in the collection C. If any element is not an element of a ring or if the elements lie in no common ring an error is raised.

gap> RingWithOne( [ 4, 6 ] );  
<ring-with-one, with 2 generators>

RingWithOneByGenerators returns the ring-with-one generated by the elements in the collection coll, i. e., the closure of coll under addition, multiplication, taking additive inverses, and taking the identity of an element.

GeneratorsOfRingWithOne returns a list of elements such that the ring R is the closure of these elements under addition, multiplication, taking additive inverses, and taking the identity element One( R ).

R itself need not be known to be a ring-with-one.

gap> R:= RingWithOne( [ 4, 6 ] ); 
<ring-with-one, with 2 generators>
gap> GeneratorsOfRingWithOne( R );
[ 4, 6 ]

returns the ring with one with parent R generated by the elements in gens. When the second form, SubringNC is used, it is not checked whether all elements in gens lie in R.

gap> R:= SubringWithOne( Integers, [ 4, 6 ] );
<ring-with-one, with 2 generators>
gap> Parent( R );
Integers

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