sentences:  Let "m" be the ideal ("x"), and let "n" be the ideal generated by "x"–1 and all the elements "z""i".These are both maximal ideals of "R", with residue fields isomorphic to "k".The local ring "R""m" is a regular local ring of dimension 1 (the proof of this uses the fact that "z" and "x" are algebraically independent) and the local ring "R""n" is a regular Noetherian local ring of dimension 2.