What is the best CoCoA-Syntax (CoCoA 5.2) for $T:=S/J$, where the base ring $S$ is itself a previously defined quotient ring $S:=R/I$ and $R:=\mathbb{Q}[x,y,z]$ is a polynomial ring and $I$ and $J$ ideals?
The obvious T:=S/J does not work.
If $J$ is an ideal in $S$ it says:
Elements must be in the same ring; mixed ring operations are forbidden.If $J$ is an ideal in $R$ it says:
Incompatible ring and ideal: ideal is not in the given ring.
I was thinking of getting the base ring of $S$ and the ideal defining $S$ and do base change of the ideal $J$ to $R$ and then computing T:=R/(I+J), which should work if the necessary methods exist, but I am wondering if there is a more elegant solution.
The CoCoA manual doesn't seem to cover this, the only thing about quotient rings I could find is: http://cocoa.dima.unige.it/download/CoCoAManual/html/partQuotientRings.html
Full code example of what I would want to work, and what works (analogously) in Macaulay2 for example:
R::=QQ[x,y,z];
Use R;
I:=ideal(x^3+y^3+z^3);
S:=R/I;
Use S;
J:=ideal(x^4+y^2*z^2);
T:=S/J;
Ooooops!! Congratulations! you found a bug in CoCoALib! ;-)
I fixed it: soon available in CoCoA-5.2.2. (if you need it urgently I can send you a pre-release: which OS?)