Using More Than One Ring

2.2.14 Using More Than One Ring

In CoCoA, every calculation takes place in a current or active ring. Ring-dependent objects defined by the user such as polynomials, ideals, and modules are automatically labeled by the current ring. Objects that do not depend essentially on the ring, e.g., lists or matrices of integers, do not get labeled.

CoCoA automatically starts with the ring R = QQ[x,y,z]. The following example illustrates setting up a ring with the construction ::= and changing rings with the command Use . One may temporarily change rings with the command Using . The example assumes that you do not already have a ring with identifier S.

Example

  Use R ::= QQ[x,y];  -- declare and use a ring R
  F := (x+y)^3;
  F;
x^3 + 3x^2y + 3xy^2 + y^3
-------------------------------
  M := [1,"test",2];
  S ::= QQ[x,y,z,a,b];   -- declare a ring S with indeterminates x,y,z,a,b
  Use S;            -- switch to the ring S
  F;  -- F is labeled by ring R
R :: x^3 + 3x^2y + 3xy^2 + y^3
-------------------------------
  M;  -- this list is not labeled R since its elements are not
      -- ring dependent (e.g., "1" is considered a separate integer, not
      -- part of the ring R)
[1, "test", 2]
-------------------------------
  F := Ideal(a^2+b^2);  -- change the definition of F
  Use R;  -- switch back to R
  F;   -- the old F no longer exists
S :: Ideal(a^2 + b^2)
-------------------------------
  GBasis(F);  -- built in functions automatically recognize the ring
[S :: a^2 + b^2]
-------------------------------