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

- Usage:
    `(F,G) = ringFromFractions(H,f)`
- Inputs:
  - H, a matrix, a one row matrix over a ring $R$
  - f, a ring element,
- Optional inputs:
  - Variable => a symbol, 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, a ring map, $R \rightarrow S$, where $S$ is the extension ring of $R$ generated by the fractions $1/f H$
  - G, a ring map, $frac S \rightarrow frac R$, 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 a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/IntegralClosure.m2:1984:0.