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presentationRing -- returns the presentation ring of a subring

Synopsis

Description

Given a subring $S$ of a quotient ring $Q$, the presentationRing $P$ is a polynomial ring with the same coefficient ring as Q and with one variable for each generator of S. There is a natural map from $P$ to $S$ that sends each variable to its corresponding generator. Elements of the presentationRing represent polynomial combinations of generators. Evaluating a polynomial combination of generators is equal to applying this map. Therefore, $S$ is naturally isomorphic to the quotient of $P$ by the kernel of the this map.

The presentationRing naturally arises when using RingElement // Subring, which takes an element of a subring and expresses it as a polynomial combination of its generators.

Subrings include the field presentationMap, which provides a map from the presentationRing to the ambient(Subring) ring.

i1 : R = ZZ/2[x,y];
 
i2 : Q = R / ideal(x + y^5);
 
i3 : S = subring {x+y, x*y, x*y^2};
 
i4 : f = x^2*y^3 + x^4 + y^4;
 
i5 : f % S

o5 = 0

o5 : Q
 
i6 : g = f // S

      4
o6 = p  + p p
      0    1 2

     ZZ
o6 : --[p ..p ]
      2  0   2
 
i7 : S#"presentationMap" g == f

o7 = true

See also

Ways to use presentationRing :

For the programmer

The object presentationRing is a method function.