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ideal(Ring) -- get the defining ideal

Description

A quotient ring is a the quotient of its ambient ring by its defining ideal. Other rings have no ambient ring, and the defining ideal is its zero ideal. 
i1 : S = ZZ/2[x,y,z];
 
i2 : ideal S

o2 = ideal ()

o2 : Ideal of S
 
i3 : R = S/(y^2-x*z,x^2*y-z^2)

o3 = R

o3 : QuotientRing
 
i4 : ideal R

             2         2     2
o4 = ideal (y  + x*z, x y + z )

o4 : Ideal of S
 
i5 : T = R/(x^3-y*z)

o5 = T

o5 : QuotientRing
 
i6 : ideal T

            3
o6 = ideal(x  + y*z)

o6 : Ideal of R
 
i7 : ambient T

o7 = R

o7 : QuotientRing
 
i8 : sing = singularLocus T

o8 = sing

o8 : QuotientRing
 
i9 : ideal sing

             3         2         2     2   2   3              4   2    2  
o9 = ideal (x  + y*z, y  + x*z, x y + z , z , x  + y*z, x*z, x , x y, x z,
     ------------------------------------------------------------------------
      3
     x )

o9 : Ideal of S
 
i10 : ambient sing

o10 = S

o10 : PolynomialRing

See also

Ways to use this method:

  • ideal(QuotientRing)
  • ideal(Ring) -- get the defining ideal

The source of this document is in Macaulay2Doc/functions/ideal-doc.m2:135:0.