The command IsPrimitive(T, R) returns true if T is primitive; false otherwise.

If the regular chain T is primitive then the ideal generated by T equals the saturated ideal $\mathrm{sat}\left(T\right)$ of T.

For the definition of primitivity and the algorithm, see the paper "When does T equal sat(T)?" by Francois Lemaire, Marc Moreno Maza, Wei Pan and Yuzhen Xie, 2008.

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$$\mathrm{with}\left(\mathrm{ChainTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\,u\,v\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$T≔\mathrm{Empty}\left(R\right)$

$T≔\mathrm{regular\_chain}$

$T≔\mathrm{Chain}\left(\left[uy+v\,xy-1\right]\,T\,R\right)$

$T≔\mathrm{regular\_chain}$

$\mathrm{Equations}\left(T\,R\right)$

$\left[yx-1\,uy+v\right]$

$\mathrm{IsPrimitive}\left(T\,R\right)$

$\mathrm{true}$

$U≔\mathrm{Empty}\left(R\right)$

$U≔\mathrm{regular\_chain}$

$U≔\mathrm{Chain}\left(\left[uy+v\,vx+u\right]\,U\,R\right)$

$U≔\mathrm{regular\_chain}$

$\mathrm{Equations}\left(U\,R\right)$

$\left[vx+u\,uy+v\right]$

$\mathrm{IsPrimitive}\left(U\,R\right)$

$\mathrm{false}$

Lemaire, F.; Moreno Maza, M.; Pan, W.; and Xie, Y. "When does (T) equal Sat(T)?." Proc. ISSAC 2008. Linz, 2008.