Ring / Ideal -- quotient ring

Ring / Ideal -- quotient ring

Synopsis:

Usage: S = R/I

Operator: / -- a binary operator, usually used for division

Input:

R, an instance of class Ring.

I, an instance of class Ideal: an ideal of R

Output:

S, an instance of class QuotientRing: the quotient ring

The names of the variables are assigned values in the new quotient ring by automatically running use R, unless R has a name, or one of the rings R is a quotient ring of has a name. See: use.

Warning: quotient rings are bulky objects, because they contain a Groebner basis for their ideals, so only quotients of ZZ are remembered forever. Typically the ring created by R/I will be a brand new ring, and its elements will be incompatible with the elements of previously created quotient rings for the same ideal.

i1 : ZZ/2 === ZZ/(4,6)



o1 = true

i2 : R = ZZ/101[t]



o2 = R



o2 : PolynomialRing

i3 : R/t === R/t



o3 = false

Code:

     -- ../../../Macaulay2/m2/quotring.m2:139-145
     Ring / Ideal := QuotientRing => (R,I) -> (
          if ring I =!= R then error "expected ideal of the same ring";
          if I == 0 then return R;
          if R === ZZZ then return ZZZquotient(R,I);
          if R === ZZ then return ZZquotient(R,I);
          error "can't form quotient of this ring";
          )