61.1 Free Magma Rings

is a free magma ring over the ring R, free on the magma M.

is the group ring of the group G, over the ring R.

A domain lies in the category IsFreeMagmaRing if it has been constructed as a free magma ring. In particular, if D lies in this category then the operations LeftActingDomain (see LeftActingDomain) and UnderlyingMagma (see UnderlyingMagma) are applicable to D, and yield the ring R and the magma M such that D is the magma ring RM.

So being a magma ring in GAP includes the knowledge of the ring and the magma. Note that a magma ring RM may abstractly be generated as a magma ring by a magma different from the underlying magma M. For example, the group ring of the dihedral group of order 8 over the field with 3 elements is also spanned by a quaternion group of order 8 over the same field.

gap> d8:= DihedralGroup( 8 );
<pc group of size 8 with 3 generators>
gap> rm:= FreeMagmaRing( GF(3), d8 );
<algebra-with-one over GF(3), with 3 generators>
gap> emb:= Embedding( d8, rm );;
gap> gens:= List( GeneratorsOfGroup( d8 ), x -> x^emb );;
gap> x1:= gens[1] + gens[2];;
gap> x2:= ( gens[1] - gens[2] ) * gens[3];;
gap> x3:= gens[1] * gens[2] * ( One( rm ) - gens[3] );;
gap> g1:= x1 - x2 + x3;;
gap> g2:= x1 + x2;;
gap> q8:= Group( g1, g2 );;
gap> Size( q8 );
8
gap> ForAny( [ d8, q8 ], IsAbelian );
false
gap> List( [ d8, q8 ], g -> Number( AsList( g ), x -> Order( x ) = 2 ) );
[ 5, 1 ]
gap> Dimension( Subspace( rm, q8 ) );
8

A group ring is a magma ring where the underlying magma is a group.

is the augmentation ideal of the group ring RG, i.e., the kernel of the trivial representation of RG.

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