# Algorithm to decide if ideal is principal

Suppose $R = \mathbb{Q}[x_1, ..., x_n]/I$, and $J \subset R$ is a given height one ideal. Is there a quick algorithm one could write to determine if $J$ is a principal ideal or necessarily not principal? Is it not possible to do this with Groebner bases?

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Various computer algebra systems have the ability to trim generators of an ideal (or module). I'm not sure exactly what this does though. math.uiuc.edu/Macaulay2/doc/Macaulay2-1.4/share/doc/Macaulay2/… –  Karl Schwede Aug 11 at 15:21