61 Magma Rings

Given a magma M then the free magma ring (or magma ring for short) RM of M over a ring-with-one R is the set of finite sums å_{i Î I} r_i m_i with r_i Î R, and m_i Î M. With the obvious addition and R-action from the left, RM is a free R-module with R-basis M, and with the usual convolution product, RM is a ring.

Typical examples of free magma rings are

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(multivariate) polynomial rings (see Polynomial Rings), where the magma is a free abelian monoid generated by the indeterminates,

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group rings (see IsGroupRing), where the magma is a group,

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Laurent polynomial rings, which are group rings of the free abelian groups generated by the indeterminates,

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free algebras and free associative algebras, with or without one, where the magma is a free magma or a free semigroup, or a free magma-with-one or a free monoid, respectively.

The coefficient ring R and the magma M cannot be regarded as subsets of RM, hence the natural embeddings of R and M into RM must be handled via explicit embedding maps (see Natural Embeddings related to Magma Rings). Note that in a magma ring, the addition af elements is in general different from an addition that may be defined already for the elements of the magma; for example, the addition in the group ring of a matrix group does in general not coincide with the addition of matrices. Consider the following example.

gap> a:= Algebra( GF(2), [ [ [ Z(2) ] ] ] );;  Size( a );
2
gap> rm:= FreeMagmaRing( GF(2), a );;
gap> emb:= Embedding( a, rm );;
gap> z:= Zero( a );;  o:= One( a );;
gap> imz:= z ^ emb;  IsZero( imz );
(Z(2)^0)*[ [ 0*Z(2) ] ]
false
gap> im1:= ( z + o ) ^ emb;
(Z(2)^0)*[ [ Z(2)^0 ] ]
gap> im2:= z ^ emb + o ^ emb;
(Z(2)^0)*[ [ 0*Z(2) ] ]+(Z(2)^0)*[ [ Z(2)^0 ] ]
gap> im1 = im2;
false

Sections

1. Free Magma Rings
2. Elements of Free Magma Rings
3. Natural Embeddings related to Magma Rings
4. Magma Rings modulo Relations
5. Magma Rings modulo the Span of a Zero Element
6. Technical Details about the Implementation of Magma Rings

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