# ringFromFractions -- find presentation for f.g. ring

## Synopsis

• Usage:
(F,G) = ringFromFractions(H,f)
• Inputs:
• H, , a one row matrix over a ring R
• f,
• Optional inputs:
• Index => ... (missing documentation),
• Variable => ... (missing documentation),
• Verbosity => ... (missing documentation),
• Outputs:
• F, , R →S, where S is the extension ring of R generated by the fractions 1/f H
• G, , frac S →frac R, the fractions

## Description

Serious restriction: It is assumed that this ring R[1/f H] is an endomorphism ring of an ideal in R. This means that the Groebner basis, in a product order, will have lead terms all quadratic monomials in the new variables, together with other elements which are degree 0 or 1 in the new variables.

 ```i1 : R = QQ[x,y]/(y^2-x^3) o1 = R o1 : QuotientRing``` ```i2 : H = (y * ideal(x,y)) : ideal(x,y) 2 o2 = ideal (y, x ) o2 : Ideal of R``` `i3 : (F,G) = ringFromFractions(((gens H)_{1}), H_0);` ```i4 : S = target F o4 = S o4 : QuotientRing``` ```i5 : F o5 = map(S,R,{x, y}) o5 : RingMap S <--- R``` ```i6 : G y o6 = map(frac R,frac S,{-, x, y}) x o6 : RingMap frac R <--- frac S```

## Ways to use ringFromFractions :

• ringFromFractions(Matrix,RingElement)