# 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:
• Variable => , default value "w", name of symbol used for new variables
• Index => an integer, default value 0, the starting index for new variables
• Verbosity => an integer, default value 0, values up to 6 are implemented. Larger values show more output.
• Outputs:
• F, , $R \rightarrow S$, where $S$ is the extension ring of $R$ generated by the fractions $1/f H$
• G, , $frac S \rightarrow frac R$, 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)"

## For the programmer

The object ringFromFractions is .