Theorem 12.1.3 of Vakil's Foundations of Algebraic Geometry is that (x, z) is not a principal ideal in k[x,y,z] / (xy - z^2). The proof is to look at the ideal (x,y,z), which is a singular point. Therefore, the dimension of the Zariski tangent space at (x,y,z) is 3. However, if we quotient by (x,z), then the point (x,y,z) in k[x,y,z] / (xy-z^2, x,z) has dimension of the Zariski tangent space is 1. Quotienting by a principal ideal cannot make the dimension decrease by 2.

This theorem can probably be generalized to any situation a singular prime ideal contains a regular prime ideal. However, I'm having trouble writing down the exact formulation of the generalized theorem and proving it. What is the most general form of this theorem?