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SubalgebraBases :: subductionQuotientRing

subductionQuotientRing -- returns the subduction quotient ring of a subring

Synopsis

Description

Given a subring S of a quotient ring Q is a polynomial ring with same coefficient ring as Q and has one variable for each generator of S. There is a natural map from the subduction quotient ring to S that sends each variable to its corresponding generator of S. Elements of the subduction quotient ring represent polynomial combinations of generators. Evaluating a combination of generators is equal to applying the aforementioned map.

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

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 : M = map(Q, subductionQuotientRing S, gens S);

                    ZZ
o7 : RingMap Q <--- --[p ..p ]
                     2  0   2
 
i8 : M g == f

o8 = true

See also

Ways to use subductionQuotientRing :

For the programmer

The object subductionQuotientRing is a method function.