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# presentationRing -- returns the presentation ring of a subring

## Synopsis

• Usage:
presRing = presentationRing S
• Inputs:
• S, ,
• Outputs:
• presRing, , the presentation ring of S

## Description

Given $S$ of $Q$, 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 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